American law generally has different rules for use of deadly force in self-defense than for use of non-deadly force:

By and large, you can use deadly force to defend yourself only if you're being threatened with death, serious bodily injury, rape, kidnapping, or, in many states, robbery (and, in some, burglary). You can generally use nondeadly force, though, to defend yourself against any unlawful attack, and also to defend your property against theft or unlawful damage.

So you can hit someone to prevent them from punching you—but you can't shoot them, unless you're trying to prevent something much worse than just punching. (There are various twists on all this, but I'll omit them for now.)

But what if you threaten to shoot them? Is that deadly force, which can't be used unless you're trying to prevent very grave harm, or nondeadly force? In Monday's People v. Ra decision (nonprecedential), the Michigan Court of Appeals held that such threats are nondeadly force:

Our Supreme Court … has applied the term "deadly force" as defined as force used in a circumstance in which the natural, probable, and foreseeable consequence of the act is death. People v Couch (Mich. 1990). In People v Pace (Mich. App. 1980), this Court determined that a defendant's mere display of a knife during a fight, while implying a threat of violence, does not constitute deadly force…. [W]ithin the context of self-defense, "[m]erely displaying a knife implies a threat of violence, but, without more, it does not constitute deadly force." This holding, that a threat of deadly force is itself nondeadly force, is consistent with Black's Law Dictionary (11th ed), which defines nondeadly force as "1. Force that is neither intended nor likely to cause death or serious bodily harm; force intended to cause only minor bodily harm. 2. A threat of deadly force, such as displaying a knife." Similarly, the treatise LaFave & Scott, Criminal Law (2d ed), § 5.7, pp 455, states in relevant part, that "merely to threaten death or serious bodily harm, without any intention to carry out the threat, is not to use deadly force, so that one may be justified in pointing a gun at his attacker when he would not be justified pulling the trigger."

That's correct, I think, and also consistent with the way other courts treat this, to my knowledge. For more on the case, which has been quite controversial, read this David French article at National Review Online.

UPDATE: Reader Noscitur a sociis notes: For in illustration of the other position on this issue, Alaska's statutory definition of deadly force provides that "'deadly force' includes intentionally discharging or pointing a firearm in the direction of another person or in the direction in which another person is believed to be and intentionally placing another person in fear of imminent serious physical injury by means of a dangerous instrument." Alaska Stat. § 11.81.900(b)(16). The legislative commentary explains, "During the legislature's consideration of the justifiable use of force, the issue whether deadly force could be threatened in situations when its actual use would be improper was frequently discussed. Because of the possibility that such threats could tragically escalate a conflict, the legislature concluded that only peace officers making an arrest should have the authority to threaten deadly force in situations where the actual use of deadly force was not justified." Commentary and Sectional Analysis for the 1980 Amendments to Alaska's Revised Criminal Code, Senate Journal Supp. No. 44 at 20, 1980 Senate Journal 1436.

(Likewise, "Oklahoma law has long recognized that, absent a threat of felonious activity, the defense of property encompasses only the right to resist trespassers with reasonable non-deadly force and, thus, does not allow a landowner to threaten them with firearms.") I much appreciate the pointer, though the Michigan view seems to be the majority view. See also Model Penal Code § 3.11(2):

A threat to cause death or serious bodily injury, by the production of a weapon or otherwise, so long as the actor's purpose is limited to creating an apprehension that he will use deadly force if necessary, does not constitute deadly force.

For another case that takes the same view as Michigan, see State v. Williams (Me. 1981) ("'Intentionally or recklessly discharging a firearm in the direction of another person or at a moving vehicle constitutes deadly force.' Obviously, the threat of firing a gun in the direction of another person without actually doing so cannot be equated with the actual discharge of that weapon.").