I am NOT a lawyer and this is not legal advice, this is only my opinion.
Just looking at a Texas Law standpoint:
Sec. 42.092. CRUELTY TO NONLIVESTOCK ANIMALS. (a) In this section:
(2) "Animal" means a domesticated living creature, including any stray or feral cat or dog, and a wild living creature previously captured. The term does not include an uncaptured wild living creature or a livestock animal.
(8) "Torture" includes any act that causes unjustifiable pain or suffering.
(b) A person commits an offense if the person intentionally, knowingly, or recklessly:
(1) tortures an animal or in a cruel manner kills or causes serious bodily injury to an animal;
(5) transports or confines an animal in a cruel manner;
(c) An offense under Subsection (b)(3), (4), (5), (6), or (9) is a Class A misdemeanor, except that the offense is a state jail felony if the person has previously been convicted two times under this section, two times under Section 42.09, or one time under this section and one time under Section 42.09. An offense under Subsection (b)(1), (2), (7), or (8) is a state jail felony, except that the offense is a felony of the third degree if the person has previously been convicted two times under this section, two times under Section 42.09, or one time under this section and one time under Section 42.09.
So it looks like torturing a previously not captured wild animal is not crime, while torturing a previously captured animal (I'm thinking like a zoo) or a dog or cat is a crime, and a felony in fact, which is important for the next section.
Art. 14.01. OFFENSE WITHIN VIEW. (a) A peace officer or any other person, may, without a warrant, arrest an offender when the offense is committed in his presence or within his view, if the offense is one classed as a felony or as an offense against the public peace.
So a regular citizen can make a "citizen's arrest" if they witness a felony in person.
Now to use your handgun as a threat of force:
Sec. 9.04. THREATS AS JUSTIFIABLE FORCE. The threat of force is justified when the use of force is justified by this chapter. For purposes of this section, a threat to cause death or serious bodily injury by the production of a weapon or otherwise, as long as the actor's purpose is limited to creating an apprehension that he will use deadly force if necessary, does not constitute the use of deadly force.
You must be justified in using force to produce your weapon as a threat, but are you justified to use force to make an arrest as a citizen? From 9.51:
(b) A person other than a peace officer (or one acting at his direction) is justified in using force against another when and to the degree the actor reasonably believes the force is immediately necessary to make or assist in making a lawful arrest, or to prevent or assist in preventing escape after lawful arrest if, before using force, the actor manifests his purpose to and the reason for the arrest or reasonably believes his purpose and the reason are already known by or cannot reasonably be made known to the person to be arrested.
So you need to state the purpose of the arrest to the person arrested before using force, so you would not be able to use the production of a handgun as a threat until you had stated that purpose.
So in your scenario, had it happened in Texas, the torturing of the turtle sounds like it is not against the law. And before you say that the clause is written that way to protect hunting, there is an exception for that already.
I think the only way for a person to use their handgun as a threat in this scenario would be to make an arrest during a felony, since protecting animals that aren't your own isn't covered under use of force in chapter 9.
Now had they been torturing a cat or dog, it would have been a felony, at which point you could make a citizen's arrest, and you could display your handgun as a threat of force only after you told the person committing the felony why they were being arrested.
I am NOT a lawyer and this is not legal advice, this is only my opinion.