I'm British and your definition 1 of wallet is what I understand it to mean. Wallet is the only word I would use to describe such an item. It would be rare for a man to carry both. Either his wallet has a coin compartment (zipped) or he carries coins loose in his pocket. In my case, both.
A purse, in my British English, is the woman's equivalent. This is often slightly larger (too big for a pocket) and carried in a handbag, but even if it's pocket sized it would still be called a purse.
The typical usage is so gendered that the same small black leather item could easily be described as "her purse" or "his wallet". "His purse" wouldn't be used for a physical item; it sounds like an item from centuries ago, or a figurative use. "Her wallet" would also seem uncommon.
A coin purse or change purse looks like what you get for a google search on "coin purse" (no image inline as the variety is what I'm trying to show).
Your second definition of wallet (Brit: a small bag) is dated as a standalone word; travel wallet is sometimes used (mainly by manufacturers, but also if carried by someone who's also carrying something like a handbag). It may be described as a man-bag if carried by a man.