Please state in which country your phrase tends to be used, what the phrase is, and what it should be.
Example:
In America, recently came across “back-petal”, instead of back-pedal. Also, still hearing “for all intensive purposes” instead of “for all intents and purposes”.
Well I find it useful to think of “by accident” as the equivalent to “by way of an accident” the accident was the way that the thing happened; there is a causal relationship there. Compare that with “On accident”, well, what does it mean to be on an accident? It sounds like a great way to get your shoes dirty.