Sorry you got downvoted, this is valid.

What I’ve described is the law school crash course version. (I was a public defender for 4 years)

What you described is the common lay person version.

I don’t like using the premeditation language because premeditation can be less than a second. The lay understanding is that premeditation needs to be actual planning, like a heist. That is not the case.

So first and second degree are better separated through analyzing how much intentionality there was in the murder. Mens Rea, or mental state, is an important concept in criminal law.

You are correct that third degree can be intent to harm but went too far and actually killed the person. I never saw that version of third degree because it’s nearly impossible to prove that you only meant to batter/assault someone when they are dead because you beat them. Of course you give the jury the option for it, but most people will just think you meant to kill them.

Manslaughter is more than a mere accident. It’s a reasonably foreseeable accident. It sounds like I’m splitting hairs, but it is nuanced. If you slip and fall into a person and they fall in front of a train and get splattered, that’s probably not manslaughter. If you playfully push a person and they slip and fall out a window to their death, that’s pretty likely to be manslaughter, maybe more.

As a disclaimer these things do vary by state and country. I’m just explaining some things that are typical of these categories.