In the Ahmaud Arbery trial, they tried to do a citizens arrest on him when they had no proof he had committed any crime. So, they couldn't claim self defense. This is logical. But what if they had witnessed him commit a crime. The crime could be on a whole spectrum of severity from stealing a pencil to trying to kill someone and everything in between. To me it is logical that if someone was running away with a pencil and you tried to do a citizens arrest and then had to shoot and kill them in self defense, you should probably go to jail. If on the other hand they were trying to kill someone and you tried to do a citizens arrest and ended up having to shoot them, you shouldn't face consequences. But there could be so many things in between that are more grey. How does the legal system deal with this? Does it just say that self defense is justified if you're trying to stop a confirmed crime no matter how small?
Self-defence has nothing to do with whether you are performing an arrest (lawful or otherwise)
Self-defence is a plea that you used reasonable force to protect yourself, others and in some jurisdictions, property, from immediate harm. There is, as you say, a “whole spectrum” of both the perception of the threat and the force used that go into determining if the actions of the defendant amounted to self-defence or not. That’s why it’s up to the jury to decide on a case by case basis.
A person who has the power of arrest (law enforcement officers and citizens who actually witness a crime) is authorised to use reasonable force to effect that arrest.
Of course, effecting an arrest may cause a situation to escalate to the point where self-defence becomes an issue.
While the "self-defense" terminology in the question is wrong, another justification for the use of force that would otherwise constitute assault, false imprisonment, or murder, in some circumstances, is to use force necessary to effectuate a citizen's arrest to the extent one is authorized by statute, or the applicable case law, to do so.
But, this use of force must be incident to a valid arrest and must be reasonable. Deadly force in effecting an arrest, furthermore, is limited to a situation in which it is necessary to prevent a forcible felony.
So @gnasher729 is also right that in that circumstance, the use of force must be proportionate and necessary, and the question's intuition, if ill-worded, that deadly force is not proportionate or authorized to make most citizen's arrests, is also correct.
The primary Georgia statute that is relevant states:
A private person may arrest an offender if the offense is committed in his presence or within his immediate knowledge. If the offense is a felony and the offender is escaping or attempting to escape, a private person may arrest him upon reasonable and probable grounds of suspicion.
O.C.G.A. § 17-4-60 (2020).
Case law annotations fill in what the statute does not state expressly:
There was no evidence that the defendant, who murdered the victim with a rifle, was attempting to effect a valid citizen's arrest and, hence, the defendant was not entitled to an involuntary manslaughter charge. It was not reasonable for the defendant to attempt an arrest with a semi-automatic weapon which the defendant was not licensed to carry as deadly force in effecting an arrest is limited to self-defense or to a situation in which it is necessary to prevent a forcible felony.
Hayes v. State, 261 Ga. 439, 405 S.E.2d 660 (1991). The pertinent analysis in the Hayes case was as follows:
(a) Appellant Turner contends that the court erred in not charging involuntary manslaughter (OCGA § 16-5-3(b)) because he involuntarily shot the victim while attempting to effect an arrest. Also, he argues that the failure to charge was error because this was the sole defense of the defendant. Turner relies upon Griffin v. State, 154 Ga.App. 261, 267 S.E.2d 867 (1980). However, this case makes it clear that a charge on the defendant's sole defense is mandatory if there is some evidence to support the charge. In Jackson v. State, 154 Ga.App. 867, 270 S.E.2d 76 (1980), the Court of Appeals found that there was some evidence from which the jury could have found self-defense. The Court went on to hold that "[f]ailure to charge on self-defense when it constitutes the defendant's only defense is reversible error." Id. at 869, 270 S.E.2d 76. A charge on involuntary manslaughter is not warranted even if it is the sole defense if the evidence does not support the charge.
The state insists that the record reveals no evidence which would support an involuntary manslaughter charge because there is no evidence that Turner was attempting to effect a valid citizen's arrest. OCGA § 17-4-60 provides that a private citizen may make an arrest if a felony is committed in his presence or within his immediate knowledge. A private citizen may make an arrest upon reasonable and probable grounds of suspicion if the offense is a felony and the offender is escaping or trying to escape. For a citizen's arrest to be valid, the citizen must use no more force than is reasonable under the circumstances. The state contends that it would not be reasonable for Turner to attempt an arrest with an AR-15 semi-automatic weapon which he was not licensed to carry. Also, the state points out, deadly force in effecting an arrest is limited to self-defense or to a situation in which it is necessary to prevent a forcible felony. OCGA § 17-4-20. Turner's argument that he was entitled to a charge on involuntary manslaughter concedes that the force used was unlawful. However, we need not reach the question whether a citizen's arrest carried out with unlawful force can be involuntary manslaughter, for we find that there was no evidence that a citizen's arrest was justified. No felony was committed by the victim in Turner's presence or in his immediate knowledge. He had no grounds for suspicion that the victim was an escaping felon. Turner may argue that he mistakenly believed that the victim was a felon. However, there is testimony that the victim answered his command to halt with the words "It's Moonbeam, don't shoot!" We refuse to find that Turner's claim of mistaken identity and his use of deadly force mandate a charge on involuntary manslaughter.
A subsequent case held that:
Although a private person may make a citizen's arrest under O.C.G.A. § 17-4-60, only force that is reasonable under the circumstances may be used to restrain the individual arrested; an alleged assault of an individual with a baseball bat entailed unreasonable force and could not have been part of a legitimate citizen's arrest.
Carter v. State, 269 Ga. 891, 506 S.E.2d 124 (1998)(citing Hayes).
In defendant's trial on a charge of felony murder, defense counsel was not ineffective for failing to request an instruction on citizen's arrest under O.C.G.A. § 17-4-60 as the defendant used more force than was reasonable in making such an arrest when the defendant shot an intruder through the wall of a storage building.
Patel v. State, 279 Ga. 750, 620 S.E.2d 343 (2005).
Whoever arrests a person without a warrant is guilty of a tort [i.e. has civil liability in a lawsuit], unless the person can justify under one of the exceptions prescribed by law; and the burden of proof that the case lies within the exception rests upon the person making the arrest.
Piedmont Hotel Co. v. Henderson, 9 Ga. App. 672, 72 S.E. 51 (1911).
You say "if you ended up having to shoot him...". No, they didn't. He was unarmed. They were three men. If three grown up men can't overwhelm a single person, and they fear that this single person may overwhelm and hurt them, then these "men" can run away, call the real police, and let them sort it out. There was no way that the had to shoot an unarmed man, innocent or not.