I think definition of a "scrap" varies from person to person. My own definition has changed; now I consider fat quarters to be scraps! (That's not a bad thing, btw, scrappy quilts are my favorites)
I have two bins for scraps - "big scraps" and "little scraps". My big scraps are anything I can cut to a "regular" size; my little scraps are more like what people call "crumbs" I think. Ever so often I go through and chop up my "big" scraps into strips, squares and rectangles and sort those into my little drawer bins. Those go into my "scrappy" quilts. The irregular small "little scraps" get used in my "crazy" quilts.
My "big scraps" are sometimes pretty large. I like to use LOF for cutting borders and I keep my backing trimmings so I end up with a lot of pieces that are a couple yards long by anywhere from 3 to 12 inches wide - I consider those scraps unless it's a good blender or solid and there's enough to border another quilt; then I'll bundle that up separately. But if it's a specific print I don't think I'll use again anytime soon...into the scrap bin it goes! I toss fat quarters in there too, unless it's a really special fabric or something I bought for a specific project. Even half-yards will end up in there sometimes - sometimes I buy a fabric online and it's just not what I wanted once I get it. Into the scrap bin it goes!
I've yet to meet a fabric I couldn't chop up small enough to make it work in a scrappy/crazy quilt.
