The modern use? muddled as hell. Paizo's Adventure Paths didn't become a thing until D&D 3E in 1999... But AP for actual play in reviews is still present and AP for Actual Play recordings is also widely present.
3e was published in 2000, but I think the first mention of "adventure paths" as a series of adventures meant to form an entire campaign was Shackled City which was published in Dungeon under Paizo's reign in 2003-2004. There had certainly been adventure series before, but I don't think they had been so ambitious.
I don't know what about a TTRPG system makes it designed specifically for LP. Any system can be used for LP. There were systems that emphasized improv and story-focused games before LP shows and podcast really took off. Has the popularity of LP as entertainment influenced more rules-light, improv-focused games.
I think this falls under "wrenches and hammers". You
can use a wrench to drive in a nail, but it's not the optimal tool for the job. And this was understood by the Critical Role crew even before they started – IIRC, the original campaign had started in 4e at first (which is why they have the Dawn War pantheon and a goliath character), converted to Pathfinder fairly early (which is why they have a Gunslinger and a cleric of Sarenrae), but switched to 5e before going public because they thought that the lighter system would be more suitable for public consumption (I can't recall if that was something CR themselves came up with or if it was a suggestion from Felicia Day).
At its core, an Actual Play depends on the charisma and performances of the participants. And the CR crew are all professional actors, and good at their jobs (at least two of them have BAFTA awards to their name). That gives them a huge edge over many others, who are often just some nerds who decide to upload their game to YouTube. They're also gaming in person which is a big advantage – many actual plays are just talking heads on webcams because the participants are sitting in front of their computers instead of playing in person, and I think you lose a lot of energy that way. But the system definitely matters. I've listened to Glass Cannon, for example, with their actual plays based on Pathfinder adventure paths (both types of AP!), and the game system definitely alters the flow. And Glass Cannon also has some acting chops on their side, though nowhere near the CVs of Mercer & Co.
"Actual Play" is a pretty set bit of terminology, y'all. WotC will use the term "Actual Play" to refer to streamed games, but "Adventure Path" is not a term that WotC uses in any capacity (they call the large Adventure books "Campaigns", have for nigh 12 years now) .
And they are right to do so. In my eyes, a "path" is something consisting of many distinct steps. An adventure path is a path of adventures – several different adventures following on one another. So it's an appropriate term to use for Paizo's typical offerings which usually come in several volumes, but not for Wizards' books which are usually all-in-one.