The definition is somewhat arbitrary, but still has some real physical significance. In actual fact, a proton is a field, so it doesn't have sharp boundaries. But the amplitude of the field still dies off very rapidly with distance from the center, so you can pick some arbitrary small value and say "the point at which the amplitude becomes less than this value is the radius of the proton". What matters is not really the number that you get out of this, but the fact that the experimental results of measuring this value appeared to change in the presence of muons. This was a phenomenon that was not predicted by present theory, and if it had held up, would have been a major breakthrough. One of the biggest problems in physics right now is that there are no experiments (except possibly this one) whose results are at odds with the Standard Model. That makes it hard to improve the model!