Yes. Passive perception "supersedes" active perception by acting as a floor.

The party scout was correct. Passive perception does not "turn off" when you are actively searching. It only stops if you are unconscious. You always notice anything that hasn't beat your passive perception score — even if you aren't actively searching.

You can also take actions to actively search, giving you an opportunity to sense things you haven't already noticed.

For example, if your passive Wisdom (Perception) score is 15, and a monster is lurking with a Dexterity (Stealth) roll of 14, you notice them without rolling. If the monster got a 15 or higher, though, they've beaten your passive senses. Let's say that the monster has a modifier of +3 and rolled 16 on the die, for a result of 19. In this case, on your turn, you can take actions to actively search, for which the DM can call for Wisdom (Perception) rolls. If you exceed the monster's total, you've noticed it. If you get the same total or less (19, here) the situation remains the same and the monster is undetected. For that first monster — the one with a 14 total Dexterity (Stealth) check — the situation also remains the same, which is that you've already noticed them. They aren't somehow "de-noticed".

Jeremy Crawford explains in the Sage Advice section of this podcast, starting at about 15:09.

JC (at 22:16): Now, going back to passive perception... this is, as its name implies, passive. And, it's considered to be "always on", unless you're under the effect of a condition, like the unconcious condition that says you're not aware of your surroundings. That really... the practical effect of that is that basically your passive perception is shut off. Passive perception is on basically whenever you are conscious and aware. [...]

JC (at 23:09): Because it's passive, the player does not get to say they use it. This is a... this is something that people...

Interviewer: (Laughs) I'm using my passive perception right now!

JC: Yeah, no. It's always on. That's the baseline. Now, this brings up questions, because then people are saying that, well, how is it that when I make an active perception check, I might get a roll that's lower? Well, you aren't... yes, that roll is lower, but remember your passive perception is aways on. So it really represents the floor of your perception.

Interviewer: Right. That's an important distinction, though.

JC: Yes. So if you make an active perception check and you get a number that's lower than your passive perception, all that means is that you did a lousy job of this particular active search, but your passive perception is still active. You're still going to notice something that "blips" onto your passive perception radar. Really, when you make that roll, you're really rolling to see "can I get a higher number?" If you fail to, well, again, your passive perception score is still active. It is effectively creating that minimum.

Interviewer: The minimum. Yeah, I don't know if that's necessarily clear to a lot of dungeon masters out there, because they will be like, well, the opposed nature of this roll means that you were just really bad at looking, and even though the person who is sneaking up on you only got like a five, they're able to do so.

JC: Now, many of these sorts of situations would be erased if DMs just simply remembered to use the passive perception in the first place. Because honestly, if something's noticable by a person's passive perception score, they should already have noticed it. So really, the active search is trying to find something that you haven't already noticed, and your passive perception score represents what you have already noticed.

(Bold added to highlight the key points; italics intended to represent emphasis in the speech.)

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.