Time for another "what do you call this" quiz!
Through North America, I've found that the burgers from Maccas and Burger King etc are referred to by the staff as "sandwiches". In Aussie (and from what I've found, everyone not working there, they are burgers), so...
Maccas/Burger King/Hungry Jacks etc serve burgers (some sort of bread roll with a hot filling)
Sandwiches are two (or sometimes three) slices of bread with some sort of filling, either hot or cold
Rolls are similar to burgers (ie a roll, not bread), except their fillings are usually cold - eg cold BBQ chicken or ham etc as apposed to a hot chicken fillet or a meat patty.
So, how do you define a burger, a sandwich and a roll?
I'm from New England.
A burger is a bun with a hamburger in it. But it's the patty in the middle that's the key element in determining what to call it. I've also seen veggie burgers, tuna burgers, salmon burgers, etc.
A sandwich is two pieces of bread with meat/cheese/veg between them, hot or cold.
Then things start to get complicated. We don't use "roll" the way you do. What you describe as a "roll" I'd call a "sandwich on a bun" or "sandwich on a roll" or even just a "sandwich." And if you put the filling into a pita, I'd still call that a sandwich on general principle.
The use of "sandwich" for burgers is, in my experience, pretty much limited to restaurants.
Then we have wraps--sandwich filling put on a piece of flat bread and rolled up.