While I (mostly) don't find alignment useful and therefore didn't vote in the poll, I wouldn't use either of the OP's definitions. They rely too much on character behavior or beliefs. Alignment is cosmic—it's either allegiance to a big, universal "side", or some inherent quality in a being's spiritual nature.
Lawful is either personal allegiance (whether consciously given or not) to the cosmic concept of Order (Moorcockian Lawfulness); or it's an inherent tie to the mundane world of human beings (and their societies and religions), as opposed to the Chaotic world of eldritch, fae, or monstrous things and Powers (Andersonian Lawfulness).