I believe the term came into existence as the former and the latter came to be because people get lazy about pronunciations over time. It's like the saying "turn over a new lease" which for some people has become "turn over a new leaf" for the same reason.
The expression “turn over a new leaf” is the correct one. It comes from when a page of a book was called a leaf, so you can see how turning to a new page can be a metaphor for making a change. “Turn over a new lease” doesn’t really make much sense, certainly not as much as “turn over a new leaf”.
You’re right that expressions get changed according to what people think they hear, especially when people haven’t seen the expression in print, or don’t think about what the expression means, or both.