I figured it out. Other faucets, you are correct, the H or C pops out and the screw is underneath but this one is different. The other day the kitchen faucet, single handle, came loose and wouldn't shut off. I had to push down on the handle to turn it off. Then I noticed the base was loose so I started turning it and sure enough, it tightened down. Swinging the faucet must loosen the faucet.
I went back to the vanity faucet and started turning the band at the base of the faucet. It was tight but turning eventually allowed the handle to come off. On the inside was a phillips head screw. I tightened it and replaced the faucet handle by screwing it on; voila!
Posting pics for future reference.
Rich & Yvonne
2006 Safari SE -Dora-
2004 4Runner SE