Re: vapor barrier
The wrong way would be to put a vapor barrier on the garage side ( between the drywall and insulation ).
Forget about the vapor barrier it's really not needed in your case.
Providing the subfloor is sheathing like plywood or OSB these materials are considered an acceptable vapor retarder. If there is a feeling to need more vapor retarder then use faced batts with the facing against the floor above.
The main thing ---- aside from moving your plumbing further into insulation ---- ensure you seal any penetration from the outside allowing cold moving air within the ceiling space which is a major contributer for cold rooms above.
Another area to consider is making sure that all penetrations from the garage into the rooms above are also sealed. Things like electrical , plumbing , cable TV line , etc. need to be sealed not only to prevent cold drafts but preventing carbon monoxide getting into the rooms above.
In this case you might install Tyvec or similar housewrap ( making sure to tape all seams ) over the insulation before the drywall. This will add an air/moisture barrier for the insulation but will allow things to breath.
Drywall is the fire barrier of choice mainly because it's realatively inexpensive and easy to install than some other materials like plaster or stucco.
When the drywall is installed make sure to use at least two coats of joint compound ----- use a primer / sealer and then a quality interior / exterior paint.
Hopefully this helps.
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