What are people's definitions of an "L3"?

Particularly, is it an architectural classification (whether or not there's some kind of intermediary between it and L1), or a security classification (can the attacker steal? can they freeze funds? can they steal/freeze but only at high cost?)

I think anything any construction that has some state and a consensus mechanism is a layer. To be an L2 to you have to follow L1, to be an L3 you have to follow L2. A proof aggregator might or might not have a consensus mechanism, depending on this it will either be a layer or not.