it's a little more complicated than that...the guys to go to on grip training are the diesel crew guys (www.dieselcrew.com
, work safe, maybe not family safe) and www.ironmind.com
(work/family safe), who also sell grip training equipment.
grip strength is split into different hand positions. open hand strength, crushing grip strength, pinch strength, wrist strength, and extensor strength.
the "vertical" grip strength you're talking about would most likely be crushing strength. "think of shaking hands," say the IronMind guys. this will give you strong thumbs, which you also develop in pinch grip exercises.
"horizontal" grip strength (farmers walks, heavy deadlifts) would be open hand strength. your thumb is less involved (unless it's a thick handled lift), but your forearm flexors get the brunt of the load.