Gradient ascent?

Or gradient descent if you mentally negate the number in question. It's the same thing.

I think hill-climbing rejects updates that would worsen the objective while gradient descent usually doesn‘t

No you don't. Go in the upward direction implies don't go in the downward direction

I don't agree that that wording is as precise as you think it is. if you move in the direction where the signal goes up, when it stops going up, you are still moving in the same direction. Move "so long as the signal goes up" would tell you to stop, but it wouldn't tell you the direction, is the type of clarity I'm looking for.