How do I finish my partially finished floor?
It is no problem and probably a good idea to use thinner to wipe up the stained poly. I would think the tricky part is going to be the tendency for the grain to get raised (rough) on the bare stained part. I think you will have to buff again but I might put 2 coats of poly down first. Then buff with a fine screen >=120 and put a final coat.

Be sure to turn off the pilot lights and watch out for anything that could flash that vapor. 

I'd give the stain overnight to dry or at least til it is dry to the touch otherwise the stain will leak into the poly and muddy up the finish.

I've switched to water-based poly. Its expensive but it dries real fast and it now outperforms oil based. No companies are putting research money into VOC stuff and they suck for the environment.