Is paper or water the stationary phase in paper chromatography?
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.