Is there a difference between 'drill-mount' and 'rimless'? These fulfill some of the criteria, but I fail to find such with a nice, large, preferably cup-shaped, lens.

it's the same thing, but lens shape is not dictated by the frame. you're likely to find pictures of rimless glasses with small lenses because small lenses are popular now, but you can choose any shape you want.

I'll see what I can dig up, thank you.