A lot of the time, "breaking the API" means "breaking the response that comes back", which in turn means that the client doesn't know what to do with the respon

A lot of the time, "breaking the API" means "breaking the response that comes back", which in turn means that the client doesn't know what to do with the response (can't parse it, missing fields, different schema etc.). If I change my API so that "POST /widgets" does something different on the backend but returns the same response, then from a client compatibility perspective this isn't a breaking change.