the reason they are named differently and are notated differently is that they serve different functions. they're more or less homophones.
or, perhaps to keep it within the artistic sphere, they're like https://en.wikipedia.org/wiki/Checker_shadow_illusion and other "same color" illusions -- they are technically the same, but taken in context they signify different things.
you would build different chords around them, you would play different melodies around them, etc. in other words, it's not just when writing them out in english that we treat those two intervals differently -- we treat them differently while using them during music
you are, of course, correct that many very competent musicians would not correctly name this distinction using the official theory terms. but that doesn't mean that they don't understand the distinction when using them in musical contexts, or that the distinction is not meaningful. plenty of professionals are experts at something without being able to describe it perfectly in words
or, perhaps to keep it within the artistic sphere, they're like https://en.wikipedia.org/wiki/Checker_shadow_illusion and other "same color" illusions -- they are technically the same, but taken in context they signify different things.
you would build different chords around them, you would play different melodies around them, etc. in other words, it's not just when writing them out in english that we treat those two intervals differently -- we treat them differently while using them during music
you are, of course, correct that many very competent musicians would not correctly name this distinction using the official theory terms. but that doesn't mean that they don't understand the distinction when using them in musical contexts, or that the distinction is not meaningful. plenty of professionals are experts at something without being able to describe it perfectly in words