that's 100% false, at least your first sentence (your second sentence is partly true). it does not matter what the charge is. once an arrest is made, the case belongs to the state. the state decides whether or not to file charges.
now, in that scenario if you punch me in the face, and the police show up, they can ask me if i want to press charges and if i say no, they can decide not to arrest me, but that is 100% the state's decision. the state is free to follow my wishes or ignore them.
now, i've heard some states have a stupid rule where if the police are called out to a domestic violence situation, they have to arrest someone. but, in practice they don't always follow this and even if they arrest someone, that's not the same as charging someone.
but in any case, whether to press charges is always the state's decision. they can sort of delegate it to the victim if they want, but they make the ultimate decision.