If I'm understanding this correctly, capacitors are placed in series with the lamp on the hot side to accomplish lamp light output reduction or current draw from the ballast?
In this case you can not separate the capacitor from the rest, it is just a part of the ballast. The magnetic part won't work without the capacitor at all. Without the capacitor it may ignite the lamp, but then the current may be either too high (so overheat it, frying the coils), or too low to underdrive the lamp (that used to be an intentional design feature to protect the magnetic part from overheating and so even reduce the risk of fire in case the capacitor fails short circuit; but it means higher cost and higher losses).
In order to work, the capacitance really has to match what the ballast is designed for.
Even when the HX vs CWA ballasts looks very similar (and they are even made using common component types, like bulk core, winding bobbins,...), they work in a complete different way and they do not work in the place of the other one. The main difference is mainly in the magnetic shunt (a magnetic insert between the primary and secondary coils): With the CWA it is designed to partially (softly) saturate at the rated current (to provide, in cooperation with the series capacitor, the required current regulation) during normal operation, in the HX transformer the shunt is designed to never saturate during normal operation (it forms the only ballasting impedance), so usually it either has larger cross section, or the ballast uses more turns on both windings.
If the capacitor is placed in parallel before the ballast, this is used for PFC?
If it is parallel to mains (could be directly, or indirectly like e.g. across 240V tabs even when the ballast is supplied from 120V). That s the case for either series reactor ballasts (99% of the 230V-area ballasts, some 240V-only and the lower wattage HPS in the US), as well as the HX (auto)transformer ballasts.
But do not mix this up with the "magnetic regulator" ballast (featuring three coils, separated by magnetic shunts). There the capacitor is connected parallel to one of the windings (the regulator one) and it's function is to form a resonance circuit with the impedances formed by the rest of the ballast. There the exact value should be used as well, otherwise the ballast just won't work at all (the capacitor is needed to form the resonator to provide sufficient OCV for the lamp start).