Reddit mentions of Advanced Engineering Electromagnetics

Beware of trying to get a "physical intuition" for the potentials- they're really just mathematical tools to aid in the solution of the real fields, E and B.

That said, sometimes the choice of vector potential is easy. If we look at their respective wave equations: ∇^2 A + β^2 A = -μJ and ∇^2 F + β^2 F = -εM, we can see that the sources for A and F are J and M, respectively. This is handy for radiation problems where you will have currents on antennas and equivalent currents on surfaces (say, for aperture radiation).

The choice isn't as obvious in waveguide analysis. I've taken graduate EM courses with both Balanis' text and Jin's. Balanis is more explicit and mathematical, where Jin tries to explain more intuitively. If you can track down a copy of Balanis, you'll see in Chapter 6 of the second edition that he's more explicit about which vector potentials you can use to solve a problem, assuming you know whether your fields are TEM, TM, or TE.

You're a bit unlucky to have been trying to get some intuition about the fields for a TEM case- since such fields can be constructed with nonzero A, F, or both. Something like a TE mode in a rectangular waveguide could only be constructed from nonzero F.

Edit: math formatting