You are correct, despite the weirdness that follows. But, as in most things "legal" it gets complicated. And, of course, I am discussing 'Murika.
Every state ("jurisdiction") has its own criminal law, and in addition there is federal criminal law that you can be subject to (dual sovereign). As a general rule, there are various statutes (or however your state calls it legislation) that criminalize certain conduct, and that discuss "deadly weapons."
The issue becomes what is a deadly weapon; in a general sense, most states define it as some variation of an object that can cause serious death or injury. This can be anything from a gun (duh) to an axe to a boat (yep) to a rake (sure) to a dog (um, okay) to your fist.
Now, there are jurisdictions that specifically define a deadly weapon as EXTERNAL to the human body.
So, aside from jokes about the inestimable Chuck Norris, what are people talking about when they say that their hands have to be registered as deadly weapons? Well, two things!
1. Pop culture canard. It's an old urban myth, that I have seen traced back to various sources (50s wrestling, pro boxers, GIs in okinawa, etc.). It's repeated in movies and TV shoes on occasion. However, it's not COMPLETELY a canard anymore. See below!!!!
2. Conflating the idea of "registering" (the myth) with the all-too-real charging (sometimes overcharging by prosecutors) under the proper state statute as a "deadly weapon" (aka, can cause serious death or injury); this has happened recently in Texas with an MMA fighter. There's a decent law review article on it here-
It does a good job explaining the differences between various states, and, more importantly, surfaced the first evidence I have ever seen of a place that requires you to register yourself! That's right. GUAM! It's unclear how serious this original effort was, as it was a recent law, it's a $5 fee, and it gives you a nice certificate for, um, framing ... but still.
Yes, you can now get registered to show that "the hands, feet, or other parts of the body are used as weapons[.]" and that you are an "an expert in the art of karate or judo, or any similar physical ar[t] in which the hands and feet are used as weapons[.]" 10 Guam Code Ann. sec. 62100, 62104.