I'm lifting the discussion here but making a concrete proposal. That discussion is also very old but no action step was taken so I believe a clean "feature request" question is better. My question is also duped to my answer there. The other answers are quite pertinent and may include better proposals, but I see this proposal as best.
It's hard for me to imagine a proper question about "climbing" that is not rock climbing, although surely something like "If a bear chases me is it safe to climb a tree?" (though an ignorant example) may plausibly come up. In that case it needs a better tag than climbing. Perhaps tree-climbing, or escape-climbing, or any more specific term from that community, but climbing is too general.
Tagging a question as rock-climbing + trad would be appropriate. It's okay to have "tag" and "subtag", but, rock-climbing swallows up way too much of climbing for this to be practical. For instance, once a tag hierarchy reaches depth 6, SE doesn't support this.
This fixes at least two problems. One, it's confusing for a user, especially a new user, as they don't know which to use and must either pick at random or educate themselves on the subtle differences between the two (the subtle differences led me to post this question, at least). Two, clicking on or searching by tag yields incomplete results, as the users were a bit random when applying the tags in the first place.
I believe you could as an option support rock-climbing requiring a subtag, but this is probably not worth the complication for this stage of outdoors.SE's development.
So it's a judgment call, and in my opinion "rock climbing" is a good, meaningful tag although questions would benefit from also being tagged "bouldering" or "top roping" or "trad", and "climbing" is a meaningless tag, because the part of climbing that means things besides rock climbing can only be muddled up with rock climbing.
Most climbing questions should be retagged to rock-climbing, or removed if rock-climbing is already present. Things like ice-climbing and tree-climbing (to run away from land mammals maybe?) may need to be applied or created. This may require domain expertise or at least the ability to create tags, as my ignorance in the second category shows. This is the burnination phase (I believe...), then we ban. That's all.