Passive perception is exactly that, passive. It's what the PCs are always using when not actively searching for something and doesn't use a roll of the die.
To determine if you should use passive perception or allow a player to roll, listen to what they say their PCs are doing. If they say they are standing watch, keeping an eye out or something similar, they are actively searching so they can make a wisdom (perception) check, otherwise they are using their passive perception.
Although, it's kind of an experience call as the DM and depends on what the PCs are actually doing and whether or not the DM gives them the benefit of the doubt. In the case of the video, one guy said he was keeping a look out, thus actively searching and two of them were scouting off to the side of the road so the DM ruled they were also searching, rather than just wandering off into the bushes for no real reason, and allowed those three PCs an active wisdom (perception) check, whilst the others would be using their passive perception.
Proficiency in a skill is used when... you are proficient in that skill. The exact amount changes by level. See the chart in the game rules, but it starts at +2 and eventually goes to +6.
From your example, the fighter (with a wisdom of 12 or 13, for a +1 wisdom modifier) has a perception modifier of +3, so that example character must be trained in perception. Perception is on the fighter's list for potential proficiencies (choose two from a list), and clearly the example character did choose that (or got it from a background choice).
Passive perception is not 14, because the passive score is just 10 + your regular bonus for that skill, not 10 + your ability modifier + your regular skill bonus which already includes your ability modifier.
The dwarven cleric isn't trained in perception, but has a high wisdom, so the +3 wisdom modfier = the skill modifier too (no proficiency bonus is added) and the resulting passive score is 10 + just the ability modifer — so, also 13, but for different reasons.
In general, the proficiency modifier is used when you are trained in (or, in other words are proficient with) the skill, tool, or weapon in question.
This is explained in Chapter 7 of the freely-downloadable basic rules, and the sidebar about Hiding on page 60 explains all of this pretty nicely. In the official terminology, there does not appear to be a name for the number you write next to the skill, other than the relevant ability modifier plus bonuses and penalties. But, colloquially, I think many people will use the 3rd edition terminology and call this a "skill modifier" or "skill bonus", or specifically "perception bonus". People will say "perception skill" too, although I think that should be avoided because it could arguably mean either the bonus, the passive score, an actual roll + modifier, or even the just whether you're trained or not.
Best Answer
Passive Wisdom checks aren't mentioned anywhere in the D&D 5e rules - only Passive Perception (and maybe Passive Insight) - though passive checks of any sort can be made at the DM's discretion. And proficiency in a specific skill doesn't affect other ability checks that use the same ability.
Note that specific skill checks are generally referred to in the rules by the ability they're associated with, such as a "Dexterity (Acrobatics) check" or a "Wisdom (Perception) check".
The rules for passive checks:
Your Passive Perception = 10 + your Perception bonus
(your Perception bonus = your Wisdom modifier + your proficiency bonus)
So if you're a level 1 character with a Wisdom mod of +0 and proficiency in Perception, your Passive Perception is 12.
If your DM decided a "passive Wisdom" check was appropriate for whatever reason, then your Passive Wisdom would be 10. But I've never heard of a Passive Wisdom check.
Perception checks can alternately be called "Wisdom (Perception) checks"; this simply specifies which ability score mod is being applied. Theoretically, your DM could call for a Perception (or other) check that uses a different ability score mod than the typical Wisdom one, if that ability score would be more relevant to the specific reason the check is being made.
For instance, the DM could ask you to make an "Intelligence (Perception) check"; this would be determined by rolling 1d20 (as usual) and then adding your Intelligence modifier (and your proficiency bonus, if you're proficient in the Perception skill) to the roll. The corresponding "Passive Intelligence (Perception) check" (which would probably never be called for by the DM) would be 10 + Int mod + proficiency bonus (if proficient).
The relevant section of the rules: