Your impression is generally right. But the law views intent as just part of the story. Randomly shooting into a crowd or dropping cinder blocks over overpasses at cars (likely second degree murder) can be viewed as worse crimes than killing someone in the heat of passion (you caught your wife having sex with another dude, you went into a rage after a rough fight), for example, which would lead to a charge of voluntary manslaughter. Arguably, a person in the former situations had a lot more control over what they were doing than the person in the latter situations despite the fact that the intent is stronger in the latter situations.