I am from the Midwest and grew up saying "purse." Then I got a job working a department store handbag counter and they were adamant about saying "handbag," so now it is that or "bag" for me. Sounds nicer, I guess.
I refer to mine as a purse, but I will switch to handbag or bag if it's a better quality item. Some luxury brands just sounds funny with the word purse behind them. But in the morning when I'm racing around the house, it's my purse that I can not find. I've never asked my husband to help me find my handbag!
It's nearly always "purse." I was born and raised in So Cal and the only people I hear saying "bag" are originally from the midwest. (And it sounds like "baig.")
The word "handbag" cracks me up. I can't help thinking "handbasket," like "going to h-ll in a handbasket." Just me? Yeah, I guess so.
To me a "bag" is generally a tote or carryall of some sort - like baggage, only smaller.