As someone who mostly just coasted in my two required physics courses in college and had no interest to study it further, this isn't really _that_ much more "obviously correct" to me than the first quote about angular momentum. Having never heard the term "generator of translation" or anything like it before, I wouldn't have been able to tell if was a rigorous, well-defined term or made up pseudoscience buzzwords before reading this thread.
This is a hyper-correction; in practice physicists apply the term in places adjacent to where it should be all the time.
FLP-II-15–5 The vector potential and quantum mechanics https://www.feynmanlectures.caltech.edu/II_15.html#Ch15-S5
" Precisely this experiment has recently been done. It is a very, very difficult experiment. Because the wavelength of the electrons is so small, the apparatus must be on a tiny scale to observe the interference. The slits must be very close together, and that means that one needs an exceedingly small solenoid. It turns out that in certain circumstances, iron crystals will grow in the form of very long, microscopically thin filaments called whiskers. When these iron whiskers are magnetized they are like a tiny solenoid, and there is no field outside except near the ends. The electron interference experiment was done with such a whisker between two slits, and the predicted displacement in the pattern of electrons was observed.
In our sense then, the A-field is “real.” You may say: “But there was a magnetic field.” There was, but remember our original idea—that a field is “real” if it is what must be specified at the position of the particle in order to get the motion. The B-field in the whisker acts at a distance. If we want to describe its influence not as action-at-a-distance, we must use the vector potential. "