assault and battery are both vaguely termed legal definitions that depend on what state you’re in.
in MN, there’s no legal definition for battery, with assault being defined with multiple levels ranging from ‘mere’ threats through “poked me on the arm” to beating the shit out of someone.
In WI, where this happened, it seems they’re the opposite with the only mention of ‘assault’ be related to sexual assault. and battery being similar to MN’s assault charges. Though, I only took a cursory look through their laws.