Is a clarinet a "small" musical instrument? If it is a wood clarinet that comes apart into small sections and is carried in a case sized to those sections, it almost certainly is. if it's a metal band clarinet that is made in a single piece and requires a case 2-3 times as long, it might not be. (Actually, that probably would be too, but I can't think of a better example right now so let's say it's a really large clarinet.) So, assuming one kind of clarinet is "small" and another isn't, what is an airline supposed to say? Or what if the metal clarinet in its case fits in the overhead bins of some aircraft, but not others, or under the seat in domestic first class but not domestic economy? You are asking for an incredible, indeed an impossible, amount of detail in terms of possible combinations of instruments, aircraft, and seats - most of which will never be needed in practice, but all of which would have to be calculated if the airline's web site were to have enough information to satisfy what you seem to want.

I think you can assume that a harmonica is small, a piccolo probably is, and a piano is not. In between, ask the specific airline about the specific flight when you have that information.