>My watch takes walking, cycling and running seriously — especially outside rather than on a treadmill — but a hard session at the gym barely registers. It will count my steps for me, but I have to count my own pull-ups

The Strava of weight training (not _counting_ the pull-ups for you, but recording them, helping you build workouts, track progress, social sharing) is the well-named Hevy: https://www.hevyapp.com