Every fast food place I've been to, a "plain cheeseburger" includes cheese. However, at every high-end burger place I've been to, "plain" does NOT include cheese. So there is a somewhat standard meaning.
I have this conversation enough that I now call out "plain no cheese" and ensure "no cheese" is written on the ticket.
That's a strange source of confusion. A plain cheeseburger without cheese has a different name: a plain hamburger. I can't imagine saying "cheeseburger", no matter what qualifying words around it, and being surprised it included cheese.
I have a very funny childhood memory of being in a McDonalds with my Cub Scout chapter; one of the boys ordered a "cheeseburger with no cheese", which - of course - was delivered with cheese, and the boy's father escalated the 'situation' to management.
> Even more confusing when McDonalds calls them "burgers with cheese".
That's a fun regional difference with McDonalds it seems, we definitely have (literally) "Cheeseburger" as a independent item on the menu compared to "Hamburger" here in Spain: https://i.imgur.com/XDNuiUW.png
That's quite funny actually, Spain tends to translate everything and have everything in English, dubbed, but apparently the McDonalds Cheeseburger got to remain, and wasn't renamed to "Hamburguesa con Queso" as one would have expected :)
McDonald's main menu items aren't actually called "cheeseburgers"; they're called "burgers with cheese". To me, this reads that "cheese" is a "topping" on the burger.
Further, they _only_ showcase the "burger with cheese variant" in their combos and special. This further drives home that you should be thinking about cheese in the same way as toppings.
You're conflating two different things. A plain cheeseburger obviously will include cheese by definition. A "plain" at a "burger place" would mean a plain hamburger. Both are correct usages of the adjective "plain" because the nouns they're describing are inherently different.
I have this conversation enough that I now call out "plain no cheese" and ensure "no cheese" is written on the ticket.