I've been looking at about 10-ish sources, and half of them say that paper is the stationary phase, and the other half say that water is the stationary phase. I understand that water is on the paper through humidity, but is there a clear stationary phase, or is it both cellulose and water (if that's possible)? Does the stationary phase depend on the humidity of the room?
The stationary phase in chromatography is the one that doesn’t move according to the eyes of a macroscopic (i.e. human researcher) observer. (That complicated way to put it was to prevent anybody raising any relativism arguments.)
Obviously, the paper does not move through the water but the water does through the paper. You should discard the five-ish sources that state water be the stationary phase as they obviously have no clue about chromatography.