Yes, a m6 chord will have the same pitch set and shape as some m7b5 chord built on a different root. Does the name matter? To basic uke players, probably not; to players of other instruments, people attempting to understand harmonic structure, people attempting to improvise over the basic chords, etc., it matters a great deal. Almost all chord shapes (even the most basic) can be interpreted as different chords or chord types corresponding to different roots. In the case of extended chords (9ths, 11ths, 13ths and 6/9s, for instance), using the wrong name may result in the player playing the wrong chord, because their voicing choices may include notes that the arranger didn't intend and which clash unintentionally with the real harmony or melody—at the very least, voicing options will be unnecessarily limited, not for the best.
m7b5 aka. the half-diminished chord (also notated ø) is the diatonic chord built on the seventh major scale note (degree) [VIIø], also on the second minor scale degree [IIø]. It occurs often in the songs I play, mostly from the 20's to 40's. Of course, its use isn't limited to these two degrees, I'm just showing that it's one of the basic chords in our familiar major and minor modes, more "natural" than the dim7 often used in its place.
If you don't see this chord type often in your sources, it may because m7b5 chords are disfavored in simpler song genres (and there are probably a lot of other common "jazz" chord types you don't see). It may be because the notator doesn't really know what's going on harmonically, and so always names those chord shapes according to the m6 names he does know. It may be because the notator is dumbing things down for the masses, eliminating less familiar chord types (a self-perpetuating circle of limitation), sticking only to names players will recognize, and reducing the number of chord changes required. Nowadays, I examine the piano parts in sheet music to restore chords omitted by simplifications in the uke/guitar notation, so that's one reason I see this chord type more often.
Note that these same shapes are also used for rootless 9th chords on yet different roots. Though there are other alternatives for playing these chords that omit some other chord component instead, the rootless 9th forms are the ones you'll most commonly see in uke chord charts and sheet music or lead sheet grids. There's at least one other not so uncommon chord type that these same shapes may be used for, but I don't recall it off-hand.
We have more control over voicing than you assume from your "first position only" and "only one shape per chord" way of training yourself. One obvious way of changing the voicing is to move chords higher on the neck. Another is to change which components of a chord we play: they're not all always necessary; for instance, you can suggest a full seventh chord with just two notes: either the third and seventh (a tritone interval) or the root and seventh—not beyond ambiguity, but the surrounding context helps our interpretation. Some chords have more than four components in their full theoretical form, so that we're forced to eliminate one or more; the choice of what we eliminate and what we keep leads to more voicing options, and again context normally helps our interpretation along.
Augmented chords should be no challenge because any note in the augmented shapes can serve as their root, and there are really only two shapes you need to learn in order to play augmented chords all over the fretboard. (In fact, just one suffices, though it wouldn't fit your limitation of always being the lowest variant possible.) Move the shape up four frets and you have the same chord again in a higher voicing. You'll notice a similar situation with dim7 chords: any note in the shape might be the root, and moving it up three frets yields the same chord by name.
One reason you blank on the non-basic chord types is because you take a harder approach to learning them. Instead of learning 12 (to 100+) chords for each type, you could just memorize four shapes—one per root string—as your go-to shapes, deriving other variants "out of real-time" when situations arose calling for something more tailored. The system which you thought wouldn't be of much use to you not only can show you how to form core shapes for each type but also all their other possible variants, and helps as a reminder (or for quick re-derivation) when memory lapses, as mine frequently does.
In a recent thread on jazz ukulele, someone linked to one of Glen Rose's tutorial videos, where he's discussing how the scary "jazz" chords he's using are closely related to the shapes for more basic chords—that's basically the system in action, though he doesn't explain the precise tweaks from a more generative perspective. A chapter of Ukulele Fretboard Roadmaps also discusses a similar derivation approach, though it does it in reference mainly to three major chord shapes (an approach I find less direct, less consistent and less complete). Sadly, the book omits a thorough discussion of chord naming, so if you haven't yet cracked that code, the UFR presentation may be mostly lost on you.
The way you talk, it is so disrespectful, but I do appreciate some information you give.
Rather than you always advertising your method with so many words, we all maybe would like to hear you play with your moving style.
I myself stick to my style that is based on chord knowledge in my head and less about physical patterns. My ukulele sounds best strummed low and balanced, providing the rhythm, Like guitar, you can't almost hear it, until it is not there.
I did play my guitar with moving barre chords to cover then the keys. It is basically the only way to play accompaniment with guitar. I would not consider myself limitated like you constantly point out when attacking me, when am just liking some style better. With guitar I stuck to patterns of play instead, with accompaniment. Also I solo of course with at least 15 frets with my ukulele.
I blank now.