Should I add a vapor barrier in this situation

No, there's no need to install a vapor barrier between two conditioned spaces. The purpose of a vapor barrier is to prevent warm moist air from traveling through the insulation and condensing on the cooler side, which can easily happen in cold climates. With a conditioned space on both sides, you can allow any moisture in the air to pass through.

There's also no need for insulation except for sound proofing (fires can pass through doors, as can any heat). For sound, make sure your insulation is designed for that, and also consider a resilient channel or similar product that will keep the drywall physically separate from the studs, which is how many sounds travel through walls.

Yep, the 120 year old houses do make life more interesting (and expensive and/or colder.)

Use an air barrier that is NOT a vapor barrier. More commonly known as housewrap. Vapor moves through, but bulk air movement is reduced.

For insulating inside the floor joists, either blown-in cellulose (which supposedly has very little issue with vapor, due to having very little air movement through it) or a sprayed foam product. Getting good fill blind (without blowing out the ceiling, in the case of the spray foam) is tricky.

I presume you don't want to rip out the ceiling for better access - but you really need to think hard about getting some drywall below the XPS (or XPS and added air barrier), or you have (and have now) a major fire hazard - plastic materials exposed in a garage that's below occupied space. The plaster ceiling above it may not burn through for a good long time, but the smoke from it will do you in even if the structure is unaffected by by a fire.