Earlier quoted context omitted.
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.
You're conflating two different things. A plain cheese burger 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.
For a "cheeseburger" cheese is obviously integral. For a Big Mac, it's less clear but a "plain" Big Mac usually includes cheese.
For a fancy place's "deluxe Wagyu beef burger" that has cheese/truffles/a bunch of other stuff, a "plain" version will likely not have cheese.