Interesting discussion! Some of this might be regional even within the U.S. When I was younger, people my parents' and grandparents' ages would call them pocketbooks. When I was growing up, we called them purses. "Handbag" was rarely a term I heard, but I was, of course, familiar with it, though never used it. When I started selling vintage, I was thrown by the term "bag." I'd never heard that used in conjunction with a purse!
Like Jonathan, though, now I generically think of them all as purses, then a handbag is a bag with handles you can only carry with your hand or over your wrist, but not over your shoulder. Funny, but I tend to think of handbags as more of a smaller type, unstructured "bag," but a larger, more structured one as a pocketbook. Not sure where these connotations have come from! Probably because I associate large "day" purses with handles as being what older folks carried in the 50s & 60s, and called pocketbooks--like those big old tapestry ones with the feet on the bottom. Also because book connotes something hard sided, and bag, something soft sided. Then you have clutches--which I think of both as bags and purses.
When listing I always try to use both "purse" and "handbag" or "hand bag" in the titles, for all versions of them. Because you just don't know what someone else is going to call it! And a change purse is always to me a change purse or coin purse, never just a "purse."
Here's what I think of as a pocketbook; it's large (10" H x 12" W) and has structured sides:
A handbag, smaller and with soft sides:
A purse; this has a short-ish shoulder strap:
we have the Italian zucchinis instead of the French courgettes.
Nicole, what do you call the eggplant? An eggplant or an aubergine?