Remember, with 5e they decided to lean more on natural language than jargon, so the onus is more than ever on us to read for comprehension, rather than on then to code unambiguously.
Is it hard to understand that 'active perception' is being used to contrast making a check to which perception proficiency will apply vs resolution involving passive perception? No. So it's a perfectly reasonable way to put it.
It's less about adhering to jargon than not confusing the listeners many of whom are already confused about what "passive" actually means in reference to a passive check. I'm perfectly capable of knowing what he means. But his use of "active" in this context muddies the water.
Personally, though my thoughts on the matter have nothing to do with who said what how, but more with the probabilities involved.
A passive check - comparing a passive score to a static DC - for instance, is nonsense. Not because it parses as nonsense, but because the DM sets difficulties, so it's just the DM ruling success or failure rather than uncertainty. There's just a superfluous step of setting a DC that will mean success or failure, then 'checking' it.
We've had this discussion before. I view it now as I did then: A passive check resolves uncertainty the same as an ability check. A task (approach to a goal) is determined to be uncertain, a DC is determined, and if the task is being performed repeatedly, a passive check is used to resolve it. If it's not repeated, an ability check is used. The uncertainty determination happens "upstream." I think I also touched on this recently in the thread on Expertise as a way to resolve issues with Expertise causing a call as to uncertainty to be an auto-success. Auto-success, like uncertainty, is determined earlier in the adjudication process.
On the other extreme I don't care for contested checks at all, they get 'too swingy' - in the sense that gets used around here.
So, that leaves checks. D20 + bonuses vs a DC. Rule success or failure or call for a check. There's one place a passive score fits in that mechanism: as a DC.
So, if a slithering tracker sneaks up on a PC, it's stealth check vs a DC of his passive perception. If a spy leaves a purloined letter at a dead drop the PCs have access to, he makes his check vs their passive investigation, perhaps?
I would say an example of passive Investigation coming into play might be as an activity while delving, say, a ruin of an ancient empire. The dungeon has pictographs or the like all along its many hallways. Rather than keep watch, navigate, forage, track, draw a map, etc., a character would have the option to study these pictographs while delving to deduce valuable lore of some worth to sages (perhaps worth X gold back in town per sector of the dungeon explored). Due to the esoteric nature of the hieroglyphs, the less than ideal conditions in the dungeon, and places where the runes were defaced by monsters or faded with time, a standard investigation uncertain. (Leaving the door open for other approaches the PCs propose to be more or less certain or difficult.) Therefore, it's a DC 15 passive Intelligence (Investigation) check at a Slow pace and a DC 20 passive Intelligence (Investigation) check at a Normal pace, and an auto-fail at a Fast pace. Further, when traveling at a Slow pace, the chance of running afoul of a wandering monster increases relative to traveling at a Normal pace. As you can see here, that specific task (or
tasks, if you want to consider the tasks at a given pace to be distinct) has an uncertain outcome with the result determined downstream in the adjudication process.
Going back to passive Perception, a character engaged in studying the pictographs is not keeping watch which means he or she is automatically surprised by wandering monsters sneaking up on the party and runs afoul of traps if the character is at the front of the party when it runs into them. That is, unless the character is a ranger in favored terrain. The upside is that hopefully the character lives long enough to go back to town and sell this useful information to the sages.