So is thermal hall effect just more general
Basically yes, but they are used in different and more specific context. In general, any (charge) Hall effect will also include thermal Hall effect, as the charge carriers carry with them energy. So thermal and electric conductivity are related (Wiedemann-Franz law). So if you just take a system with a charge Hall effect and apply a temperature gradient you will get a thermal Hall effect carried at least by the same charge carriers (e.g. electrons).
The phonon Hall effect was referred to as something that happens when only phonons are available for energy transfer, and then the only measurable Hall effect is the thermal one. From a theoretical point of view, it requires some breaking of the chiral symmetry in the phonon system. As acoustic phonons do not couple directly to the magnetic field, you either need to look at optical phonons for this, or to somehow couple the acoustic phonons to some symmetry breaking term. From an experimental point of view, the signatures are very weak and measuring a thermal Hall effect is quite challenging.
Recently, other types of exclusively-thermal Hall effects which do not involve phonons came into focus, as chiral spin-liquids do not have any electric Hall signature but should show a thermal Hall signature. As this type of systems attracts a lot of attention right now, there are some recent works on thermal Hall effect regardless of phonons.
So basically, the term "phonon Hall effect" is used exclusively when analyzing transport of phonons, while "thermal Hall effect" is more general, and pertains to a larger class of systems and energy carriers.
EDIT for follow up questions:
1) at least for quantum spin-liquids, the point is that the thermal Hall effect carried by the spinons should be quantized (and fractionally so for things like Kitaev spin liquids). And this is probably the strongest signature for such a QSL. This was (supposedly) measured two years ago by Kasahara et. al., which led to a lot of interest. However, theoretical explanations of this experiment also relied on phonons to carry the energy and heat (See here and here). Experimentally, at low enough temperatures the contributions of the phonons will be negligible, while the contribution of the gapless edge mode will still be quantized. So there is a point in studying just the spinons contributions in QSL.
2) No. Wiedemann-Franz law only relates to charge carriers, as it relates amount of energy and amount of electrical charge transported. You can say that for phonons it applies in a trivial manner (you multiply the thermal conductivity by zero charge, and get the electrical conductivity). When you have both types of carriers in your system and they both contribute, you have to calculate their contributions separately. However, mostly there will be some separation of scales and one will be a correction to the other.
In addition to that, there is further complication as the quantum anomaly that gives the Quantum Hall Effect (any of the such) does not apply to the quantum thermal Hall effect, and WF law may break in such topological systems. See here for an example, and the beautiful theory paper by Mike Stone, who I think also contributes on this website.
3) See for example here, how small the phonon effect is. It is usually masked by contributions from electrons, and one needs an insulator to see it, and even then it is tiny.
4) by definition, acoustic phonons don't couple to magnetic field, and therefore the most obvious time-reversal symmetry mechanism does not effect them directly (higher order mechanisms will couple them to the magnetic field, by they will be smaller in size). Optical phonons couple to the magnetic field by they are massive so at low energy can be neglected.