I appreciate the elaboration on the example at hand!
So, in both of these cases, the DM is describing the PC’s action. This is something I specifically avoid doing. In my games, it would be
Player: “I go over to look at the vase”
DM: It looks like a regular vase.”
or
Player: “I look at the vase and the pedestal closely, trying to see it from all angles.”
DM: “Make an Investigation check.”
Sure, in the case where any physical interaction with the vase sets off the trap, if doesn’t matter if it’s done with a tool or by hand. However, had the vase been coated with contact poison, or cursed, or a mimic or something, it might have mattered.
In my case, they would have to state that because, no, I’m not going to assume they’re touching it just because they didn’t say they didn’t. Say what you do do, and I’ll resolve that action.
I’d prefer the player say what their character is doing, so I don’t have to assume either way.
One of the biggest differences is that unless an obstacle such as a trap is being totally bypassed (i.e. not interacting with the vase at all) that the risk falls on the roll of the dice and the build of the PC, not player ability. I want to level the playing field a bit because player skill makes a pretty big difference no matter what. So it doesn't really matter who adds fluff to the description of an investigation, it could be me or the player. The only thing that really matters is that the PC is investigating and whether or not they physically interact with the vase, otherwise the description isn't going to matter. I'm not going to assume physical interaction unless the player specifically says so. At that point, I don't see that it matters if the vase is a mimic or not, it's time to roll initiative.