Players Do Not Initiate Passive Checks
The operative word is passive. If a player is actively searching/examining/studying/watching, it's active, not passive. If a PC is actively using a skill, they roll for it instead of the DM using a secret passive check.
The only exception to that rule is the one you quoted: the DM can use a passive check to find the average result of doing the same thing over and over again, regardless of whether it's active or not.
In general, the point of the existence of the passive check is so that the DM can determine something when the players aren't expecting to notice anything. In those cases asking them to roll would give away that there is something, so instead D&D 5e provides DMs the passive check to avoid giving away the secret. When to use passive checks is much easier to grasp when you look at them from the perspective of the problem they were designed to solve.
Also as a consequence, PCs are rarely in situations where it makes sense for players to ask for passive checks — after all you can't know to check what you don't know is there to check! This puts the responsibility for ensuring passive checks are made when they should be on the DM's shoulders. As a useful aide for this, it helps for the DM to keep a note listing each PC's name and their most important passive checks. Just having the note can be a small spur to remind the DM that passive checks should be kept in mind during interactions with hidden things, NPCs, motives, and etc.
To answer the titular question, this means that a player never states that they're making a passive check. It's something the DM does as the situation requires.
Just ask the Player
Often times it is pretty obvious when the player will use Cutting Words or similar features. When my evil sorcerer is about to cast a powerful spell the PCs identified I ask if the Bard wants to cast cutting words.
Similarly, if a player is very low on HP, I ask. I usually wait a bit before declaring if the attack hits (I use the time where I'm explaining the action narratively to see if the party member wants to use their reaction) And yes, it is good to know what reactions are available for your party; however, you could also simply ask if anyone wants to do something after an attack roll.
Lastly, Don't be afraid to retcon the story as long as the player using their reaction made sense; you don't need to punish them unnecessarily.
Note: I removed the list of some relevant class features, since some might perceive it to be too selective and unhelpful; the edit log still carries the list if you are interested
Best Answer
You're right
... and so is your DM, technically. A 1 is the lowest you can roll on the d20 die: it is the worst possible result for that roll. But your modifiers can affect that greatly.
If your character gets a +10 by a certain level, at that point he will auto-pass all DC 11 or lower skill checks that use your skill. This is because critical failures for ability checks do not apply. Those are reserved for attack rolls and death saves.