Almost right, Nico. The modern classical terminology is actually derived from the transposition necessary based on C - composers & arrangers need to know what interval of transposition to write a part in and they’re C based. The concert flute, including Böhm, is and always has been acoustically in D, but because conventionally it was never treated as a transposing instrument (i.e. the fingering-name/notation symbol/actual note sounded are directly correlated), composers think of it as “in C”.
The trad (including military band) naming of the flat key band flutes represents their actual acoustics, their home “straight off” diatonic scale and their normal lowest note - and if you want to understand their playing properties & propensities is the most useful and most accurate; but it will mislead a composer/arranger. Said music writers won’t think, “Ah, that’s Bb in relation to D, I need to write the part a major third higher”… they’ll think it’s Bb in relation to C and only write the part a tone higher (as for Bb clarinet) - and hey presto, it won’t fit in ensemble. So said folk need to label the Bb instruments as “Ab” (etc. - all a tone lower than their diatonic home key/acoustic centre) to make the correct transposition. But that does not change the constructional or practical playing nature of the instruments!
So, both systems of nomenclature are valid, neither is “wrong”. It all depends on your perspective. If you play them, the “official” modern classical terminology (only prevalent even in classical circles from the last 1/3rd or so of the C19th, BTW) is really misleading, but if you don’t play them and are writing for them, the traditional, true pitch names are almost equally unhelpful.
For me the clincher in arguing for a sensible standardisation is that even as a C-centric classical type, it helps you to write for an instrument idiomatically if you understand well how it behaves and how its players perceive it, and the “correct” transposition name is then rather misleading. As in, the easiest key to play in on a Bb flute is (actual sound) Bb, written in D. It is not particularly easy to play in Ab (actual sound), written in C. OK, classical types work from a given premise that well trained players can perform in any and all key signatures with equal facility, whether or not some are easier or harder than others. But in reality few players of simple system flutes will be happy playing in more than 4 sharps or 3 flats (written key sigs), bearing in mind written C major is already in effect 2 flats (F# and C# flattened from the home scale to F and C).
Nor, of course, do many of these band flutes have extended foot joints - they mostly stop at the 6-finger note/written D fingering. A composer seeing the label “Ab flute” might well assume it was like a Böhm flute and write down to a low Ab (actual pitch, written C) it doesn’t have. IMO far better for these people to study the instruments a little (as they’d expect to for most other things beyond the orchestral norm) and understand their nature. If they then use a mental shorthand of “Bb flute needs Ab transposition”, fine.
Here’s a summary of band flute transposition info for composers/arrangers/copy-writers illustrating how I believe they would do best to think of the matter:
Simple System (and historic) flute nomenclature (and also that of tin/penny whistles) is based on the actual pitch sounded by the 6-finger note (usually the lowest available) and the home diatonic scale proceeding up from that by sequentially removing the fingers from the open tone-holes, the register break occurring between the 7th and the 8ve. The 6-finger note is conventionally written as D (below the treble clef stave), regardless of actual pitch sounded. Although most band flutes have a full chromatic scale available in a tessitura of at least 2.5 8ves from their fundamental (to 3rd 8ve written A or above), in key-signature terms the further away from the key of D the music is written, in general the more awkward it will be to perform. These instruments are sometimes misleadingly named by their C-centric transpositions, a tone lower than their actual home tonality.
D “concert” flute & piccolo - written at pitch, though piccolo written 8vo basso (non-transposing “C” instruments)
Bb band flute (bass or treble) - written up a major third (treble instrument written 8vo basso) (“Ab” transposition)
F band flute & piccolo - written down a minor third (picc 8vo basso) (“Eb” transposition)
Eb/D# band flute & piccolo - written down a semi-tone (picc 8vo basso) (“Db/C#” transposition)
To establish the correct transposition for a flute or whistle labelled in any key other than D, establish the actual pitch of its 6-finger note/home diatonic scale and write that note as D, making your transposition by the interval between that note and D.
Cautionary note: the above is predicated on all instruments being built at/for Concert Pitch A=440. Many (probably the majority, but by no means all) antique instruments and some modern ones made for flute bands are constructed for late C19th English High Pitch, approximately a semitone higher, c A=452-5. Many flute bands still use HP instruments and will not have alternative equipment at CP. This will not normally affect transpositions for music written solely for flute band (carry on as above), but if you are going to mix in band flutes with other instruments at CP, first ascertain what pitch standard the instruments are built for! If you want (inadvisable, really) e.g. to mix HP Bb treble (“melody”) flutes with a CP band or orchestra, you’ll have to allow for an extra semitone (written D, nominally Bb, sounds a CP B! - so transpose up a minor third), think carefully about what key signatures you are asking both parties to play in, and hope the band flutes can tune to an exact semitone sharp of CP nominal.