I agree you can't try to 3rd-degree murder someone and accidentally 1st-degree murder them, but you can try to non-fatally injure someone and, due to being mistaken about the facts, end up killing them (e.g. because you were mistaken about the effects of a poison you used). In that case, you do indeed get convicted of a lesser crime, if the court/jury believe you.

In the theft and criminal damages context, you could have an analog, though we don't currently, where being judged guilty of one of the more major theft or damages offenses (like "theft over $X") requires an intent to cause that level of theft or damages. If you meant to cause $50 in damages and actually caused $50k, that could be a lesser offense than if you meant to cause $50k in damages. You'd be found guilty of essentially "damage over $X but with intent to only cause damage under $X", the way 3rd-degree murder and manslaughter are lesser offenses due to the lack of intent to cause death, even when there was intent to violently injure non-fatally.

I'm still in law school, so I can't claim expertise yet, but my understanding is that accidentally killing someone while purposefully injuring them would be 1st-degree murder.

When you get excused is if you didn't act purposefully... e.g. if you drive your car recklessly and kill someone you'll be guilty of manslaughter. If you purposefully hit someone with your car you'll be guilty of murder even if you only meant to break a few bones.