Is saying one is using the thieves tools to disarm the trap any more specific than they are using their eyes and a good light source to search everything near the door for traps?
Well see, now you’re talking in terms of character action, which is different than “I use my perception skill.” Using your light source to examine something is at least an actual declaration of action, though I don’t really see it revealing anything the description of the environment didn’t already, unless your light hadn’t previously reached whatever you’re trying to examine.
How else would a thief check something for traps besides using their eyes?
Interacting with it physically in some way?
So do they need to mention they are checking the seams to make it clear they are finding the lever trap?
It’s not that she needed to mention checking the seams - there could have been any number of other ways she could have found the trap. But I did determine that this action did succeed in her finding it.
What do you envision in your mind when they say they use their their thieves tools to disarm the trap? Do we even know what thieves tools are in any particularly specific way that the picture everyone envisions are vaguely the same?
From Xanathar’s guide: “Thieves’ tools include a small file, a set of lock picks, a small mirror mounted on a metal handle, a set of narrow-bladed scissors, and a pair of pliers.” I imagine the character using those tools to probe the mechanism for the trap, perhaps jamming it or triggering a partial activation or something similar. A certain degree of abstraction is necessary here because neither I nor (presumably) the player know the precise mechanics of the trap or how to disarm it with those tools. The important thing is, I know generally what the character is doing within the fictional space.
If they say they use their thieves tools, but a hammer and spike or some rope would be needed do they auto-fail for not mentioning they use those too? (How do you disarm a lever when you only see the end of it in a crack using standard thieves tools?)
You’re imagining that I have decided in advance on certain actions that will be needed to disarm the trap. This is not the case, and indeed I specifically avoid doing so, because doing so is exactly what leads to the dreaded “gotcha” gameplay.
If mentioning the seam where the trap was was can give an auto success in finding it, can mentioning doing something to the lever (I duct tape it and super-glue it in place) give an auto-success on disarming it?
Well, let’s think it through. Your goal is to disable the trap. You approach is to secure the triggering mechanism in place with glue and duct tape. Can that approach succeed at accomplishing that goal? I certainly think so. Can it fail at accomplishing that goal? Sure, you could accidentally bump it too hard and actually cause the bell to ring. Does failing to accomplish the goal have a consequence? Yes, the bell ringing would be a meaningful consequence, as it would alert nearby enemies (that’s what the trap was designed to do), not to mention the time it would take to apply the tape and glue and wait for the glue to set. So, I would say this would require about 10 minutes (my standard “dungeon turn”) to attempt and require a successful Dexterity check. I’d probably go with DC 10. On a failure, you accidentally trigger the trap.
What happens if they say they're searching the seams but the trap actually a pressure plate under the carpet in front of the door --that they'd need to step on to check the seams?
I’d say something like “when you step closer to look at the seams, you feel something sink under your feet and hear a click. What do you do?” Depending on what the pressure plate triggers and what the player says they do in response, they might be able to avoid the effects of the trap without needing to make a save. For example, if they say “I hold my breath,” they would automatically succeed if the trap releases poison gas, but not if it fires poison darts. Though I would still ask them to make a Dex save in the latter case, because not getting to roll to save because you didn’t blindly guess what the trap would do correctly would suck.