Which is open to interpretation. I've told you how I rule it and why.The questions you ask are answered by the text you quoted.
So real simple real world scenario. The group comes across a chest. The rogue Sly's player says "I check it for traps". There's no particular hurry, so the few seconds to check for traps is not an issue. As a DM you know it's not trapped. Do you:
A) Ask for an investigation check even though there is no trap.
B) Tell the player "Don't bother, it's not trapped."
C) Ask for an investigation check and if it fails, make up a trap on the spot so there's a consequence to failure.
D) ???
I do A because it reflects the interaction of Sly engaging with the world. If Sly's player wants to check every 5 feet for traps or every door, we'll talk about pacing and using passive checks as a group.
What do you do and why?