Minnesota (since I happen to live there) has 5 degrees of assault:
- First Degree assault requires "bodily injury which creates a high probability of death, or which causes serious permanent disfigurement, or which causes a permanent or protracted loss or impairment of the function of any bodily member or organ or other serious bodily harm." (nb., this is the assault offense I suspect would be charged; attempted murder and other charges may be levied, too, but I'll stick to assault for ... concision?)
- Second Degree assault involves a deadly weapon and seems to require "the fracture of a body part, temporary or substantial loss or impairment to a bodily function or organ, or an injury resulting in temporary but substantial disfigurement."
- Third Degree assault involves "leads to substantial bodily harm" but doesn't involve a deadly weapon.
- Fourth Degree assault happens "when a person assaults another in the performance of their duties as a public servant" (eg., punching a police officer, even if there isn't "substantial bodily harm").
- Fifth Degree assault appears to be the catch-all for "anything else"; eg., punching someone in a barroom brawl.
Those are, of course, just the high-level overviews; there are specific statues that define some of those terms and put hard requirements to charge a specific assault as (eg.) 1st rather than 2nd degree.
Most crimes also have a "Mens Rea" component: the perpetrator needs to have a "guilty mind" - that they intentionally did something that they knew was illegal. My understanding is that this is what allows pleas of temporary insanity or charging crimes of passion as lesser offenses. As the Kyle Rittenhouse trial demonstrated, what many people see to be a clear example of a crime might not be due (at least in part) to the Mens Rea requirement and/or other mitigating factors. This is also why minors (especially young children) are treated differently: a 5-year-old barely has the concept that other people are actually people or that consequences can last past bedtime; it's incredibly unlikely that they can have the guilty mind necessary to commit most crimes.
Note that Mens Rea slightly more specifically means that "the defendant must be conscious of the “facts that make his conduct fit the definition of the offense.”". It's conceivable that someone might not know that (eg.) a pocket knife could be considered a deadly weapon, which might take 2nd degree assault off the table. Some statues don't have a Mens Rea component, some explicitly have one, and some seem to imply one; I'm honestly not sure where the various assault statues fall.
The US also has a "reasonable doubt" standard for conviction - this is supposed to be a very high bar, erring on the side of letting the guilty go free rather than incorrectly convicting the innocent.
And, the prosecutor needs to charge a specific crime. That is, they need to specifically charge "Assault in the First Degree"; they do, in some scenarios, have the ability to file multiple charges - in principle, they may be able to charge both first and second degree assault, and let the jury decide whether the actions in a particular case cross the threshold from 2nd to 1st.
So, at a minimum, the prosecutor in the Hadi Matar trial must prove, beyond a reasonable doubt:
- Hadi Matar is the person who attacked Salman Rushdie
- he did so intentionally
- he knew that doing so was illegal (or, at least, that the action was likely to cause damage, or whatever else the statue requires him to have known)
- and any other requirements of the specific allegations - was a deadly weapon involved, was there a high probability of death, was there serious permanent disfigurement, was there temporary disfigurement, was there permanent or protracted loss or impairment of the function of an organ, etc..
The "innocent until proven guilty" philosophy says that the jury (and, indeed, the criminal justice system, generally) needs to assume the answer to all of those questions is "no" unless and until the prosecution can prove the answer is "yes" beyond a reasonable doubt (with some fuzziness around setting bail and such).
Simply showing that I stabbed Frank is insufficient to prove first-degree assault. In fact, it's insufficient to prove any assault, since I might plead mitigating factors (eg., self-defense). Similarly with Hadi Matar: simply proving that he stabbed Salmon Rushdie is insufficient (though necessary) to prove assault.