The problem is compounded by the various manufactuurers and modelers wanting "smaller but larger".
O is allegedly 1:48 scale, so "hal-O would be 1:96, but it's actually 1:87.
N should be 1:192, half of what HO is supposed to be, but again it's larger, and so it goes.
The scales are more logical than it appears. In the beginning, gauge was more important than scale. Back then, trains were more toys than models. So getting the gauge standardized was more important for than scale any hope of interoperability.
0 (now "O") gauge - a German gauge naming convention - was 32mm, very close to the 1.25" adopted by Lionel and other American manufacturers. The British, in the meantime, came up with a 7mm/ft scale (1:43.5) to fit O gauge track, which is fairly close. HO became Half O at 3.5 mm/ft - which it still is today (1:87.1).
In non-metric America (this is the '30s), 1/4" per foot was deemed close enough, and became O scale - 1:48. Thus, the mismatch in American O between scale and gauge. There have been attempts to create an "American" HO at 1/8" to the foot - which would be half of American O. Europe and the metric world use the more correct 7mm/ft for O.
The Brits perpetuated this nonsense with OO/HO. Because British loading gauge is small, and tiny electric motors were not available, 4mm/ft was used for HO gauge track in Britain. When OO was brought to America, Americans would have no more of the mismatch, and insisted on a corrected gauge for OO (which probably doomed it to commerical failure, as it was too close to HO in size, but wasn't interchangeable).
S and TT were US-invented logical scales based on inches and feet, with no metric influence.
A British manufacturer, Lone Star, started making OOO equipment using 2mm/ft scale, the next logical step down from OO. I dont' recall if Lone Star actually used 9mm track, but it was at least close. At the same time ('63-'64), German firm Arnold Rapido started a new scale which also used a gauge close to 9mm. Cooler heads prevailed, and a correct scale and track gauge were standardized on, using 9mm track to represent 56.5" prototype. There still is a British 2mm/ft group of modelers, some using N gauge track. Note that other than the rather indirect connection via British OO, N has no relationship to HO.
Z was another German initiative.
trying to clear the air