The Hendrix chord is usually mentioned when a harmony contains both the major and the minor third, together with the minor seventh. Were it a major seventh, I'd say we have an augmented 9th and thus it would make the 7#9 notation seem more logical. However I tend to address the Hendrix chord 7b10. Why am I wrong?
It's because how we build chords. The most typical way to build chords is in thirds thus when we name intervals with respect to a chord we use odd intervals to represent them. Also note a 10th reduces to a 3rd which is one of the basic building blocks of chords. It doesn't make sense especially from a functional standpoint to have two thirds in a chord.
Let's look at the chord as use in Purple Haze and see why we make the distinction between the two. The song starts out with the E7#9 and it's very distinctive. Before arguing the #9/b10 semantics, let's just look at the first half of the chord E7 which is spelled:
E G# B D
A dominant 7th chord is one of the most common chords in music and we recognize it is a dissonant chord that wants to resolve. Whether it resolves or not affects how the progression sounds, but that's a different topic for a different day. Let's look at the chord the two different ways as a #9 and as a b10.
In this scenario we will call the added tone a G (b10). So we are thinking of the added tone as a third instead of a 9th. Thirds (and tenths) in general are considered very consonant would make it seem like we're not adding much dissonance to the chord itself which is already dissonant and we'd be expecting a more minor sound to the overall chord. However, when you look at the chord as a whole you have the dominant base which contains a major third and is already dissonant so adding the minor third which is a semitone away from the major third creates more dissonance and doesn't really produce the minor sound you would expect when the minor 3rd is added.
In this scenario we will call the added tone a Fx (#9). So we are thinking of the added tone as a 9th instead of a 3th. 9th in general are considered dissonant like 7ths with an alteration (# or b) make them much more dissonant. We realize the E7 is already dissonant and this tone is making a lot more dissonant and the 9th with it's alteration shows the dissonance pretty well where as the b10 covers up the idea of dissonance. It even has the bonus of being the next extension (a 9th) we would encounter in the chord.
While yes a b10 and a #9 are enharmonic equivalents, what we think about in a 10th vs a 9th is very different and the 9th better describes how the chord behaves. Calling it a 9th also as pointed out above also has the benefit of being the next extension encountered so even though it may seem odd to call a minor 3rd #9, it fits with how we build chords in third in general.
It is just convention. If you want, can find arguments for both symbols #9 or b10. For this reason I also think that arguments trying to show that one of the two is "wrong" are rather beside the point. One pragmatic reason to stick to #9 would just be because it's much more common, at least in the Anglo-Saxon world.
But to give an example for b10: I live in the Netherlands and I also studied music (jazz) here in two different schools, and it was the first time in my life that I came across the b10 notation. I found out that at least in the Netherlands, the use of b10 instead of #9 is very common, and not among people who might not know better, but among reputable music theorists. If I remember correctly, their arguments were something like:
b10 resolves downwards via b9 to the fifth of the I chord: e.g. G7(b10) - G7(b9) - Cmaj7 (notes: Bb->Ab->G); #9 would indicate that it resolves upwards, which is never the case.
b10 may come from the altered scale where it's the third note, not the second, so again b10 instead of #9.
b10 may come from the blues scale, which arguably has a minor third in it (and not an augmented second)
I hope I don't do these people injustice by wrongly remembering their reasons. But again, this is all relatively unimportant because #9 or b10 are just symbols to communicate with others, and as long as they are interpreted correctly everything is fine. In sum, you are not wrong, you just use a less conventional notation. You might want to consider moving to the Netherlands.
Whenever I suggest the "Spinning Wheel" or "Hendrix" chord sounds a lot more like a flattened third on top of a dominant 7th structure than like a sharpened anything, I am shouted down by supporters of "pile of thirds" orthodoxy. They'll let me have a C6 chord though. So the only answer I can offer is that it's a religious thing.
You are absolutely correct to think like that, the correct way to describe this chord is as ♭10 chord, not ♯9. The ♯9 is a theoretical misconception made by jazz theorists of "functional harmony". Historically this chord is written in classical sheet music as a ♭10 chord (you can check Villa-Lobos Cirando No.4 "o cravo brigou com a rosa" to prove my point). You can also read about this chord in any classical harmony book. To further prove my point, in any songbook/fakebook by Hal Leonard, Almir Chediak, Berklee press etc. You will find the minor third/tenth written in the melodies, but the chord symbol is 7♯9. the whole "we build chords over stacks of thirds" just doesn't make sense, since there are 6th chords in standard chord notation. The ♭10 misconception is a lot similar to the subV chord misconception. The SubV chord originates historically from the #6th chords (Italian, German and French sixth chords), however, for some reason they decided to treat it as a dominant 7th chord, instead of a #6 chord. The sheets and classical harmony books back up the ♯6 argument. As Laurence Payne mentioned above - it's a religious thing; people just won't admit it, even though the same guys who theorized 7th♯9 are the same ones who call the blues scale a "minor pentatonic" scale. Why not just call it "♯9(omit3) scale" then? My head hurts from trying to argue with these guys...
Andrais, even with the flat 7, I still think the effect is one of augmentation or suspension.
When I see this chord used, it is over a major sonority, usually. You can skip the #9 (with all its excitement) and play the major chord without doing violence to the harmony.
Try to substitute a minor triad only (without the major third) and it likely will not sound right, or work with the melody.
So that #9 - such an exciting, even violent sound - is properly, I think, an augmentation type accidental. Aesthetically, it feels like it is reaching up, not curving down, like a regular minor 3rd. Now that is purely subjective. We shall see if anyone has a different feeling about it.
Of course, in the end you are right - it certainly is a flat 10, but decades of usage always count for something when it comes to these things. I saw an informal lyric/chord chart showing an A# chord not too long ago. The head spins. While your fingers reach for the Bb automatically, A# (while technically correct) stops you in your tracks just when you need to go, go go.