Okay, let me take another stab at it. I think we've gotten silly - I blame myself for injecting that coffee commercial - and are making it harder than it needs to be.
A "brim" is the uppermost edge of any container that can be filled. Avangi seems to think the container needs to be round, but I disagree. "Fill to the brim" isn't an idiomatic expression, although it can be used figuratively (he was brimful of merriment). "Fill to the brim" keeps coming up in this discussion, because it's about the only way we use this word (except for hats!)
When we're not speaking figuratively of brims, we're filling a cup, or something similar to a cup, with liquid. When something is "brimful" it has been filled to the point where liquid is actually slopping over the edge, or creating the surface tension "bulge" that Avangi mentioned just before it slops over. Unless the container has fairly vertical sides, it can't be filled in that way. So, platters and trays don't have brims.
However---unless we're discussing hats!---a "brim" isn't really an actual part of the cup - it's just that point at which the cup is full. You wouldn't say "a fly is perched on the brim of my glass, which is brimful of milk" even though the fly and the milk were at the same spot. You'd say the fly was on the "edge" or "rim" of the glass.
I love these soupbowls with the wide blue rim. I would love them more if they were filled to the brim with your delicious cucumber soup.