in this discussion, I've noted several usages of "by" and
"on" We began with "on accident" and "by accident"

Other common (or regional) uses of these prepositions noted are: "by intent" , "by mistake", "on account",
"waiting on" (in Indiana I heard "are you waiting on me?) , there is travel "by car", "by plane", by train" , "on foot"
and "on purpose".

Let me add one more not mentioned yet that's related to "by accident", it's "by chance".