In my view, the first two are incomplete action declarations, because they don’t state a goal. My guideline is that I must be able to identify both a goal (what you want to accomplish) and an approach (what you are doing to try and accomplish it.) When in doubt, a statement that takes the form of “I want to _____ by _____” almost always contains all the necessary information. That’s why “I try to pick the lock by using my thieves’ tools” works - I know what you want (to get the lock open) and how you plan to try and make it happen (picking it with your thieves’ tools), so we gucci. “I intimidate the orc” only tells me that you want to get the orc to do something, and you want to apply your Intimidation proficiency if a roll is required, but it doesn’t tell me what you want to get him to do or how. “I threaten the orc” is the same statement, just with a different synonym swapped in for intimidate. The third works, because it tells me what you want the orc to do (run) and how you try to get him to do it (say you will do horrible things to him if he doesn’t).
The first one wouldn’t work at my table because it didn’t tell me how your character is interacting with the desk. I would have to make a lot of assumptions about what you think “use investigate” means, opening the door for misunderstanding or hard feelings if I assume something incorrectly, especially if it leads to an outcome that is harmful to your character. I also have to assume what you’re even looking for, though I don’t feel too unjustified in assuming “anything of interest.” With the second, I have a bit better idea what your character was doing “going through the desk” is a little vague for my liking, but I understand it to be common shorthand for “open all the drawers and look through whatever’s in them,” so that’s fine. I still don’t know what you’re trying to find, but again, I don’t feel too bad in assuming “anything of interest.” I would probably accept that action declaration, but only because asking for more specificity would probably be more trouble than it’s worth since this one is close enough in most respects. The third one is a complete action declaration including both goal and approach, though I would prefer you state your goal confidently instead of asking a question. Replace “is there anything important?” with “I’m looking for anything important” or something similar and you’ve got a beautiful action declaration.
