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The simple answer to the question you asked is that they are not mutually exclusive. Self-defense and “castle doctrine” are defenses. A person can be charged and tried for murder, and one or both of those can be their defense. But shooting someone in self defense does not guarantee immunity from a charge or trial. In the first place, you need to show that it was indeed justifiable self-defense.

As a source for this answer, see Texas state law library. https://guides.sll.texas.gov/gun-laws/stand-your-ground

That site itself says that the laws are complicated and refers readers to “plain English” from which I selected https://www.bhwlawfirm.com/deadly-force-self-defense-in-texas/

For self defense, the site says:

Texas law provides for a justifiable defense at trial when using deadly force if the person claiming self defense:
Reasonably believed the deadly force was immediately necessary;
Had a legal right to be on the property;
Did not provoke the person against whom deadly force was used; and
Was not engaged in criminal activity at the time the deadly force was used.

For protection of property, it says

Under Texas Penal Code §9.42, a person may use deadly force against another to protect land or property if:
He is the owner of the land;
He reasonably believes using the force is immediately necessary to prevent arson, burglary, or robbery; and
He reasonably believes that the land or property cannot be protected or recovered by any other means.

OK! To summarize the story linked by the OP, homeowner hears and then finds a man outside breaking into homeowner’s shed. He confronts intruder who then moves toward homeowner with a pickaxe. Homeowner shoots and intruder runs off. let us agree that the first shot was allowed under Texas law, preventing a robbery and perhaps an attack with a pickaxe. The homeowner says that he then shot again “into the night.” At this point he is shooting a fleeing person. We can even leave out all of the irregularities once he calls 911 two hours later to report an invasion in progress even though the intruder was dead.

In any case, there is also the questions are:

1. Is the homeowner’s version of events true in the first place?


2. If we accept everything he said, was the shot the killed the man justifiable under Texas law?

These are for the prosecutor to decide if it is worth trying and the jury to decide.

Back to your question of how can they charge him if he had a right to stand his ground?

Further, even if a person has a justification for using force, he may still be arrested and face trial.

Self defense is a defense against a murder charge, not a get out of jail free card.

A charge of murder means that the government are accusing someone of intentionally and unlawfully killing someone else. A charge is an accusation, not a conviction. A conviction is when you are found guilty of a crime. You aren't a murderer until you've been convicted, since innocent until proven guilty is a principle of law.

Self defence is a defence to murder: if you can prove you acted in self defence, then the accusation of murder will fail. You will not have committed a murder, and you will be found innocent for the charge of murder, though you might be charged and convicted for a lesser crime such as manslaughter.

It is true: legally, they are mutually exclusive. No one can be convicted (found guilty) for murder when they have successfully raised the defence of having acted in self defence.

What is happening in the news article is that someone was charged with murder, and that person may or may not raise self defence as a defence. Once the trial concludes, the jury will decide if self defence is successfully raised or not.

Murder is almost universally defined as deliberate homicide, or in some equivalent way.

"Self defence against an attack" is an affirmative defence to a charge of causing physical harm, be it battery or murder.

It is to explicitly acknowledge that the deliberate homicide occurred, but the circumstances of that homicide justify it in some way that should excuse the defendant from any criminal penalty.

Where murder charges are actually laid against a defendant who posits acting in self defence, it is the belief of those laying such charges (District Attorney, Crown Prosecution Service, etc.) that the circumstances do not justify homicide, either because lethal force is excessive to resolving the situation or because no attack was actually occurring.

A court can acknowledge that the accused was defending themself, but decide they acted too violently under the circumstances and is guilty of murder by virtue of excessive or unlawful self-defense. This is called "imperfect self-defense".

Imperfect self-defense is a common law doctrine recognized by some jurisdictions whereby a defendant may mitigate punishment or sentencing imposed for a crime involving the use of deadly force by claiming, as a partial affirmative defense, the honest but unreasonable belief that the actions were necessary to counter an attack. Not all jurisdictions accept imperfect self-defense as a basis to reduce a murder charge.
_{Wikipedia article on Imperfect self defense}

Summary

Whether "murder" and "self-defense" are mutually exclusive depends on the jurisdiction and how the self-defense argument is mounted.

In Texas, imperfect self-defense can either mitigate the severity of the crime (reducing a murder charge to manslaughter) or it can mitigate the punishment after the defendant has been found guilty of murder (reducing a death penalty to a life sentence, e.g.). So self-defense and murder are not always mutually exclusive.

In the U.K., imperfect self-defense is a mitigating factor in what is still called "murder".

In Maryland, imperfect self-defense and murder are mutually exclusive. (But I'm not sure my one citation is proof enough.)

Texas-Specific Rules

The Texas statutes under Title 5 (Offenses Against the Person), Chapter 19 (Criminal Homocide) set out the definitions for "Murder", "Capital Murder", "Manslaughter", and "Negligent Homocide". We'll just look at "Murder" and "Manslaughter" as being relevant here.

Sec. 19.02. MURDER.

…

(b) A person commits an offense if he:

⇒(1) intentionally or knowingly causes the death of an individual;

⇒(2) intends to cause serious bodily injury and commits an act clearly dangerous to human life that causes the death of an individual

…

Sec. 19.04. MANSLAUGHTER.

(a) A person commits an offense if he recklessly causes the death of an individual.

So in their basic definitions, it seems that deliberately shooting someone would qualify as "murder" regardless of self-defense being applied, while "manslaughter" would apply if the shooter was just aiming wildly in the dark trying to scare off the shootee or something similar.

The Texas statutes under Title 2 (General Principles Of Criminal Responsibility), Chapter 9 (Justification Excluding Criminal Responsibility), Subchapter C (Protection Of Persons) provides defense-related definitions. Sections 9.31, 9.32, and 9.33 describe "Self Defense" (non lethal), "Deadly Force In Defense Of Person", and "Defense Of Third Person" (lethal or non-lethal), respectively.

Under those definitions, "self defense" explicitly refers to using non-lethal force against someone, while murder requires the use of deadly force. This means "self defense" (in this very narrow definition) and "murder" are mutually exclusive. However, it's clear from reading excerpts from pretty much any murder case that the term "self defense" is also used for lethal force cases in practice, so this logic doesn't hold for real-world cases.

Of note is that neither the homocide statutes nor the justification statutes mention anything about imperfect self-defense. Either you were defending yourself legally, or it was murder. However, there is common law related to the subject.

Appellant therefore was not entitled to have the self-defense issue submitted to the jury at the guilt-or-innocence phase of his trial. Nevertheless, his imperfect self-defense claim warranted submission of the provocation issue at the penalty phase.
_{Charles Evans v. State of Texas at the Court of Criminal Appeals of Texas (1980) via Justia}

.

Under the common law doctrine of imperfect self-defense, if a defendant merely intended to assault the victim and ended up killing him because of escalating violence that he did not intend, the actor would be acquitted of murder and convicted of another offense, depending on his culpability.
_{Sergio Huerta v. State of Texas at the Criminal District Court No. 2 (2008) via Justia}

.

As explained below, imperfect self-defense … [is a] defensive issue that may be relevant in evaluating a claim of self-defense in certain cases.
_{Erick Santos-Valdez v. State of Texas at the Fourteenth Court of Appeals (2014) via Justia}

In the 1980 case, imperfect self-defense was only useful in determining the sentence (specifically, death penalty vs. life sentence in this case), and not in determining the charges. In this case he was convicted of capital murder, but imperfect self-defense was at play in overturning his death penalty. So murder and imperfect self-defense were not mutually exclusive. (Appellant robbed a store and claims he only shot the manager because the manager shot first. Clearly, the appellant had no right to self-defense so the murder conviction was upheld, but because he was shooting defensively, it was a mitigating factor in the murder sentence.)

In the 2008 case, imperfect self-defense might have been used to reduce the charge from murder to manslaughter. So in that application, murder and self-defense were mutually exclusive. (Appellant shot in the air during a vehicle robbery to scare the victim. Victim shot appellant. Appellant claims he only walked to the vehicle and unloaded a magazine into the victim because he was surprised by being shot. Appeals court disagreed.)

In the 2014 case, imperfect self-defense is irrelevant to the case, but it's worth noting that the concept is still valid as of five years ago. (Appellant shot a man who was grappling him through the window of his car, after the appellant's friend tried to rob the man. The appeal summary isn't very specific on why this was murder, since it's only dealing with technicalities.)

Other-Than-Texas Rules

The UK still considers it murder, but allows mitigating the minimum term of the prison sentence, if self-defense was somewhat involved:

As murder is such a serious crime, the approach to sentencing for this offence is set out in law.

…

Mitigating factors are things that may reduce the minimum term. These include:

…

• the fact that the offender acted to any extent in self-defence or in fear of violence
_{Sentencing Council for England and Wales}

The state of Maryland explicitly considers "murder" to include malice, while a legal defense of "imperfect self defense" explicitly requires proof there wasn't malice. As such, "self defense" and "murder" are mutually exclusive under legal definitions, while "self defense" and "manslaughter" are not.

Initially, we note that the difference between murder and manslaughter is the presence or absence of malice.

…

Imperfect self defense, by contrast, is not a complete defense. Its chief characteristic is that it operates to negate malice, an element the State must prove to establish murder. As a result, the successful invocation of this doctrine does not completely exonerate the defendant, but mitigates murder to voluntary manslaughter.
_{State of Maryland v. Melvin Faulkner at the Maryland Court of Appeals (1984) via Google Scholar}