HID and fluorescent ballasts really aren't all that different, and do all the same jobs, in many of the same ways. A key difference, which can cause some confusion, is packaging. In North America, fluorescent ballasts are usually packaged in potted "cans" with all of their components (except starters on preheat) contained inside. HID ballasts, on the other hand, often have all of their components separate. The parts you have will be the ballast core/coil, an ignitor (for pulse start systems) and a capacitor (for high power factor systems).
HPS lights (and pulse-start metal halide) require a very high voltage (up to 4000 volts) to start, but than run on a very low voltage (around 55 volts for low wattage HPS lamps). This is done using an ignitor. There are three styles of ignitors, external 2 wire,
external 3 wire and internal. It sounds like your old HPS ballasts had internal ignitors, where the ignitor is packaged within the ballast core/coil, either underneath the core wrappings
(Advance), or in a box attached to the core
(Universal). Your new ballasts sound like they have external ignitors. 2-wire style produce the high voltage pulse internally, and are wired in series between the lamp and ballast (as well as neutral). 3-wire style ignitors generate a pulse, than use the windings of the ballast to build it to the necessary voltage to start the lamp. Those wire to the connected between the lamp and ballast, the neutral connection and a tap in the ballast. For best performance, those are matched to the ballast.