If climbing stairs is your exercise, then use "Outdoor Walk" on the Apple Watch

I have had success getting the Apple Watch to count stairs -- but NOT with the generic "other" app. Instead, use the "Outside Walk" option. This measures elevation and distance. I think the problem is the "other" option, which I labeled "Stairs," does NOT count elevation.

When I used "Outside Walk," the Apple Watch correctly counted my stairs (120 or 5 flights of 24 stairs). In addition, it gave me other measurements (see pictures), pulse, and all the other TMI you expect.