Thank you! That's exactly the kind of expert advice I needed. The front face of the tub deck is also tile. My instincts were to grout, but I didn't want to make a mistake and be sorry later.

If the wall tile butts into the floor tile, then I'll color-match the wall grout. If the floor tile butts into the wall tile...well, I can't recall ever doing that. I always tile the floor before setting the bottom course of wall tile. Mongo

Usually the caulk matches the grout. It would be grouted in that area, but the change in planes (the countertop is perpendicular to the backsplash) means that there will be more movement because of the way houses move. If it was grouted, the grout would crack because it is not flexible. Caulk is flexible, so it can withstand the normal movement of a house better - less chance of cracking, though eventually it may also crack and need to be replaced - one of the normal maintenance things one has to do around the house.

Run a bead of paintable caulk between the top and wall and paint to match. And tell those guys they're a bunch of hillbillies.