| HX means high leakage inductance (auto-) transformer.
The lamp needs to be fed from something approaching a constant current source, so a device which in ideal form feeding constant current and the voltage is then determined by the load.
And of course the available energy source is just a constant voltage source (so maintaining giiven voltage, while the current is then given by the load). So you need a device converting the constant voltage mains nature towards the constant current type the lamp needs. And that is called the ballast
Of course an ideal constant current source is not feasible, but a voltage source, significantly higher than the load voltage, with some series impedance is quite good approximation. As that voltage source appear on the terminals when there is no load connected, it is called an "open circuit voltage" (so OCV) and it is the second most important parameter of a ballast, after the rated output current.
As all the conversion has to be done with high efficiency, the impedance is formed by either inductive or capacitive reactance.
With 230V mains the mains voltage is sufficient to act as the OCV, so such lamp ballast could be just a series inductor. But that works only in 230V countries.
In the 120V areas the mains voltage is not sufficient, so it requires to be boosted to the required 220..300V OCV and then the current limited by the inductance.
And the HX transformer is just a single device combining the two functions above: Boosting the voltage, as well as providing the required series inductance. Compare to a standard transformer this is done so, the primary and secondary are separated and a magnetic shunt is put between these allowing the primary magnetic flux to return without reaching the secondary, when the secondary is loaded. The parameters of that shunt (the reluctance,...), together with the number of turns then dictate the output current. The OCV is then the turn's ratio as with any other transformer, as without the load the easies path for the magnetic flux is through the secondary.
The other, more advanced type of ballast is CWA, a.k.a. "Constant wattage autotransformer" (well, the name is misleading, as it is a better constant current source). It consist of two parts: Electromagnetic assembly (the transformer part) and a capacitor in series with the secondary. There the transformer part provides the required voltage boost, but the impedance is the series combination of the capacitor and leakage inductance.
Because capacitive reactance has an opposite phase to inductive one, these two actually subtract from each other. When the capacitive is the larger one (that is the case for CWA), decreasing the inductance means the overall impedance will increase.
Now the indyuctances have one habit: If you exceed some current limit, part of the core (here the shunt) get saturated and so the inductance reduces above certain current.
Now if you combine this inductance reduction into the overall impedance, you get a significant feature: Increasing the current means the inductance drops, so increasing the overall impedance, so in fact going against the current rise. This feature of the CWA then really stabilizes the arc current and makes it way less dependent on mainly the mains voltage fluctuation. If that is combined with a lamp of constant arc voltage (that were the MV's), you get a setup maintaining constant lamp power even with varying mains voltage. And this property then coined the name of this ballast type.
As the MV lamp has the voltage drop rather independent on the temperature (as all the mercury is evaporated, so the gas density can not change anymore), this type of ballast gives the best performance (goof arc maintenance, supressing mains voltage tolerance and fluctuations), so it was the most common type for MV's (or it's insulated secondary variant called "CWI"; well, when the cost was more important, the simpler HX were used as well)
However with newer lamps, designed to operate with the fill not completely evaporated there is one problem: If the lamp temperature happens to become higher, it means higher vapor pressure, that means higher voltage drop and that means with really a perfect current source higher power delivered to the lamp, that leads to further increase in temperature. This feedback then may be prone to instabilities and runaway, so with these lamps the imperfect nature of a "more conventional" HX ballast to thermally stabilize the setup (there the increased arc voltage gives less room for the inductance, so the current drops; plus the partial saturation makes that effect even stronger by increasing the inductance at lower current).
Therefore for the pulse start MH's the ballasts are of the HX type.
As the MV is rather immune towards the ballast type, it works on a HX as well, it s only a bit more sensitive for the mains voltage fluctuations.
Now why a 100W MH ballast to use with a 125W MV? Well, as I wrote, the primary ballast parameter is the lamp current, secondary the OCV. The lamp power rating is used as a lamp designator, but it is by itself irrelevant to the ballast vs lamp interface - there we need to match the current and OCV.
The 125W MV is rated for 1.15A current, providing about 120V arc drop, with minimum OCV 200V.
The 100W MH is 1.2A rated with a 85V arc voltage drop, requiring an OCV of 200..250V.
Now if you take such inductive ballast and recalculate what will happen when the arc voltage would be the 120V, you get current arount 1.15A.
Viola, that happens to match the MV requirements pretty well. Now the lower current means slightly lower losses in a HX ballast (all currents are in fixed ratios), so no problem there either. Of course, the transferred power to the lamp will be around the 125W and not just the 100W as with the 85V MH arc, but that is not important for the ballast at all.
So it means the 125W MV is compatible with the 100W MH ballast...
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