(girlfriend here)
OK, so after doing some more research I think I have figured this out. The old engine was a dual circuit, so there was a DC wire that charged the battery (red wire on the pictured harness) and an AC wire that powered the lights (orange wire). The new engine is a 9 amp regulated system, so it has one DC wire coming off the alternator to charge the battery, but no separate headlight wire. The adapter I bought takes that one wire and connects it onto the mower's wiring harness to charge both the battery (red) and power the lights (orange). There is no separate switch on the mower to turn the lights off and on, so I guess this means with the new engine the power to the lights is always on. The B&S wiring diagrams seem to confirm this, as the wiring diagram for a dual circuit system does not have a light switch, while the wiring diagram for the 9 amp regulated system does have a light switch.
So...if I'm understanding this correctly, it looks like my options are to take out the light bulbs, disconnect the spliced wire on the adapter so the orange headlight wire no longer gets juice, install a light switch so the headlights can be turned on or off at will, or disconnect the battery when the mower is not in use. For the time being I think we are just going to disconnect the battery until we figure out what to do in terms of snipping wires and such.