Part of the reason for insisting on specificity of action and wanting to know more about what a player is trying to accomplish is to make certain success and failure are meaningful. If a player wants to know how strong is a troll in comparison to us a successful Nature check will reveal how potent a creature it is. If they want to know about weaknesses a successful check will let them know that it is weak to fire. I want to avoid the scenario where a successful check results in a massive info dump that does not meaningfully answer the player's question.
I'm not sure if you're framing this in a "step on up" context or more generally. I don't play much of the former and so will try and speak to the latter.
When I actually reflect on how action resolution unfolds in my game I can see that specificity of action is a very variable thing.
For instance, in Classic Traveller when the PCs were trying to jury-rig communicators to increase their broadcast range the player narrated inversions of flux capacitors and connections to ATV power sources - but this was really all just colour, as none of as has any working conception of how the Traveller technology works (they can make interstellar journeys but can't built communicators as small or powerful as 90s-era mobile phones?). Key to the framing of the check was the skill-set the PCs brought to bear, their possession of the requisite skills, and (when it came to setting a DC) me being guided by some pretty generic examples found in the rulebooks. As far as consequences were concerned when they succeeded but not super-well, I made something up about overloading the device so they got the longer range for the scene (where they wanted it) but then the communicator was burned out (so they lost that bit of their gear).
In the same Traveller campaign, when a PC was wrestling with a NPC for control of the latter's sub-machine gun we had a detailed sense - at the table - as to who had a grip on the gun at what point and who was able to fire it. This created some implications as to (say) who was on top and had what sort of hold on the other, but that stuff wasn't spelled out in detail. It was left implicit.
Reflecting on these examples, but many others, I suspect that I give
player's intention for the outcome of the PC's action a pretty high degree of importance in adjudicating outcomes. Specificity can be a means to that end, and also can be relevant to framing, but that depends quite a bit on the system (how much specificity is needed for framing - eg consider combat positioning in 4e compared to Prince Valiant) and on how the system handles outcomes (in Burning Wheel injuries need to be to a location; in Prince Valiant it's mostly just a bit of narration over the top of temporarily-depleted Brawn).
And back to 4e: this is where I have the most experience of "monster knowledge" checks, and I tend to just follow the rules and dump information based on the degree of success. I think that 4e combat works best when the players have a general sense of what their opponents can do but there is the odd surprise combined with fear of what it can do.
I've only used a handful of monsters in BW: zombies, a mummy, a sphinx and a dark naga. The approach to learning stuff about them is very different from 4e, as it turns on the use of Wises and failure can always yield unhappy consequences. (I can't remember the details, but I think at one stage a failed attempt to learn more about the mummy - by reading symbols on a scrap of bandage it had left behind - established a curse in the fiction.)