I can at least say with certainty that the "300a" label on the adapter is nothing but wasted ink. The adapter has no electronics, thus it has no limit of its own. CCS, with my EVHub adapter, has gone 500a on a 500a station multiple times - and the official Tesla adapter (which I'll have very soon!) is no different - but likely a little more robust (thus stays cooler, thus stays faster, longer - as the car and station adjust charging rate based on port temperature as well).
The communication is also maddeningly complex - pilot amperage is something AC J1772 does, but DC just signals "I'd like to speak digitally, please" in its amperage signal, then all the real logic happens in the digital realm. The car and station exchanges bits back and forth hundreds of times a second, pitter-patter-chitter-chattering with complexity so deep there's entire teams dedicated to implementing it.
With DC charging (Supercharging, CCS, CHAdeMO, whatever), the station charges the battery directly - thus the need for such ... intense communication. It's not feeding a fixed voltage to the car (like 500v, or anything). The car just opens up its battery to the cable and says "feed me, very specifically". If the car or the station don't like what each other are doing, *snap*, they can cut the whole conversation off. And that's a theory as to what's happening with that station in Alaska (or generally any other time you're at a CCS station that just... randomly stops): one side or the other didn't like something, so *snap*, it stops.
ninja last-second edit to add:
That's not to say (by "the station charges the battery") that the station has the "final say" in the matter. Both sides agree, constantly, on what they want to see. There is no forcing, there can be no "unwanted touching". Both the station and the car kinda have a "final say" in everything. Station has limits, car has limits. Station has needs... car has needs. Thus, tons of communication. More than in some marriages.