I suggest chooseing depending on how hard and long you want to work on this. Regardless wherther you strip or sand,chemically clean before starting and again before applying first coat of new finish. Most people will recommend taking every square inch of finish off and sanding every dent and scratch smooth then filling grain regardless how long it take's to finish. Interesting thing is that when it's THEIR stairs less than 1 out of 10 follow their own advice. Instead of going to all that trouble,they lightly sand and apply a new coat,as witnessed by present condition of your stairs.

Polyurethane is best removed with a solvent gel type of stain like Strypeeze. I've used it on antique oak furniture and it works well. It does have pretty strong solvent fumes so be prepared for that.

Start on a small area, paint on, let it sit a few minutes and try scraping with a putty knife. If the finish doesn't come off, wait longer. After scraping most of the finish off, use fine steel wool wetted with stripper, or a toothbrush, to get in the nooks and crannies. Have plenty of rags at hand to wipe. After the finish is all off, do a final rinse/wipe with mineral spirits to get the last residues off. It's tedious for trim and carvings but you can't haul your staircase to a dip-and-strip place.

I'm not that fond of sanding it off because it can damage the curves and edges.