I personally don't have a "default", I just make a call based on the situation.
In general, it depends on how noisy the invisible creature is relative to the environment and how much it interacts with the environment. But it all comes back to making a judgement call, what could be perceived? I take into allowance the fact that some PCs are far more perceptive than I am, but there are still limits.
- Flying creatures in an open area are difficult to detect, particularly smaller creatures (no detectable down-draft). An invisible imp flying above the market is going to be nearly impossible to detect unless they disturb some birds, fly through smoke or similar. Of course it will change if the imp is screaming epithets or throwing things.
- In situations that are relatively quiet, I'll probably require a stealth check. Sneaking past the guard (before combat) is an example. I may adjust and give advantage/disadvantage based on environmental factors.
- In combat it depends. Mass battle chaos or 4 PCs vs 1 opponent in a quiet cave?
- Environment matters. On a city street? Middle of the day with people running and screaming because a balrog just gated in or middle of the night? Storm and wind meaning you have to yell to be heard or calm and quiet?
- Is the ground wet, covered in snow or is there another reason to leave tracks such as walking under cover during a rainstorm?
But for me, I don't have a huge issue in part because I don't have 5 minute work days and I've never given out items that grant greater invisibility equivalent. Spell slots (even at higher levels) are pretty precious and greater invisibility only lasts a minute. If the PCs can take advantage of this now and then, fantastic. It's more of an issue with creatures that have greater invisibility, but that just gives me options to throw different challenges. The scouting invisible imp or sprite familiar is probably the biggest headache I ever have. Then again, lanterns of revealing aren't that expensive to have scattered around in highly secure areas. YMMV.
I don't see the OP's scenario being all that much different from what happened in a recent game. The PCs were placed in magical darkness by monsters that could see in the dark (orc blades of something or other I think). I rolled straight dex checks for the orcs vs passive perception; there was a decent chance the PCs would know where the orcs were but not guaranteed. This was in the abandoned part of the city, outdoors so not dead silent and a mix of hard and soft ground. Depending on if the PCs beat the dex check and by how much they either had a pretty good idea of location or just a general direction.