This question already has an answer here:
I was wondering about the foundations of the way we name intervals.
For example, the interval between C and G is a fifth because there are five notes from C to G. But it's a common mistake of the person learning intervals for the first time to count four: one step from C to D, one from D to E, one from E to F and one from F to G. This is what I call "distance", because one intuitively counts the steps.
It seems to me that both approaches are valid if one was to rebuild the principles of music theory. So the question is, why is it the way it is? I think that advantages and disadvantages are worth mentioning but I also think that the truth behind this must be on historical reasons.
Edit: to sum up, I see there are (were) two options for naming intervals. One was counting note by note, and the other, counting the steps. My question is how and when the first one became the standard.
Please take into account that why this option was “chosen“ is relevant but not the main interest, so in my opinion this shouldn't be considered a duplicate of Why aren't intervals zero indexed. It's rather a variant, with a different point of view.
Regarding motivation for the question: I saw little children getting confused because they counted four steps from C to G, and I realized that it was perfectly natural to make that interpretation. As children, they don't have their minds pre-programmed as grown ups do. We consider that calling that interval a fifth is natural exactly because of that: we got used to it.