The good news: light water stains on oak are often not nearly as dramatic as they look. The wood is usually just swollen on the surface, not actually damaged. Sanding is really the last resort, and I wouldn't consider it unless nothing else works.
Start with the iron trick, but do it really carefully - too much heat can actually make the stains worse or even discolor the wood. Lay a damp (not wet!) cotton cloth on the spot, then run the iron over it on low heat. That often helps get the moisture back out and relax the wood. If that doesn't work, try one of the tips with vinegar or toothpaste - the important thing is to wipe everything dry with a towel immediately afterward and don't scrub like crazy. By the way, with oak flooring, aggressive rubbing can create light scratches that become a problem.
Maybe give it two or three days - sometimes the stains just disappear on their own once the moisture escapes from the wood. If something's still visible after that, you could also try working over it with a parquet oil (specifically for oak), which can sometimes visually even out small discolorations.