This is a surprisingly hard question to answer! An orthodox point of view says "diameter" is just a word for a physical quantity which can be measured by exper

This is a surprisingly hard question to answer! An orthodox point of view says "diameter" is just a word for a physical quantity which can be measured by experiment and plugged into a formula to predict the result of other experiments. In my personal opinion, theoretical physics is not currently capable of providing a satisfactory answer, but if we knew more about the relationship between quantum field theory and general relativity it would probably be possible to say more. There are plenty of ways to wave your hands about it, and I have been doing this for a long time and could give 100 different explanations. At the end of the day what do I see in my head when I think about the "diameter" of the proton? Basically a probabilistic soup which is confined to a small area. The "wave function" of a quantum particle can tell you where it is but is not the same as the particle itself. Standard QM assumes particles are infinitesimal points, in QFT this leads to infinities which can be resolved if you assume particles have a "smeared out" radius. This "smeared out" radius is what the proton diameter should be related to when the more complete quantum and gravity theory is discovered.