Marina has her facts perfectly right: "couldn't care less" is the older expression, and it also a more logical fit between the syntax and the intended meaning. Nonetheless, the conclusion she draws that "couldn't care less" is the correct form and "could care less" is therefore incorrect because it "really" means you "could care some", irks me, as it is based on at least one false assumption about the way language works, and perhaps more than one.
The short story is that "could care less" an idiom and that in the U.S., it appears to be significantly more common in conversational English than "couldn't care less." It's also empirically false that it's used disproportionately by teenagers or the uneducated. It is attested among a wide range of ages and social classes. (See this post in particular for support for this claim.)
The fact that "could care less" is an idiom means that the argument from logic is irrelevant. Idioms are "chunks" of language. They develop particular meanings independent of what a literal interpretation might suggest. The meaning of "kick the bucket" cannot be predicted from the meaning of the phrase that we would derive if we treated this as a new expression and interpreted it from its basic composition. Similarly, the lack of a logically necessary negation does not affect how people actually understand "could care less," although like all idioms, it is likely to confuse non-native speakers who have never encountered the expression before. In fact, if you want to use "could care less" in its non-idiomatic, compositional sense, you must use contrastive stress, and you probably need some relevant context as well: "It's true, I could care less, but I'm still not very interested." If you're going to say that "could care less" really means "do care," and not what everyone who uses it thinks it means, logical consistency demands we apply the same rule to every idiom: "kick the bucket" really means "swing your foot so that it comes into contact with a pail"; to "get someone's goat" really means to retrieve their farm animal; unloosen really means to tie.
The other possible assumption that may be present is that "couldn't care less" is right because it comes first. That privileges etymology over current usage. But the assumption that the original, or at least older, meaning of a word or phrase determines its meaning for all time is patently false, an example of the etymological fallacy. No one today would think (I hope), that it is improper to call a wooden house dilapidated by virtue of the fact that this word comes from the Latin root for stone, lapid-. Yet this same assumption is behind the silly notion that decimate must always mean "to kill one out of ten," and pretty much every other usage gripe that is supported by an appeal to how the word or phrase was supposedly used in the distant past. (I say "supposedly" because often these complaints are based on false etymologies, but my point holds even when the etymology is correct.)
So whenever I see etymology used as to draw prescriptive conclusions, I get crotchety. Stories about words' origins are fine as far as they go, and I probably find them just as interesting as the average subscriber to Marina's YouTube channel, but to me, the interesting part of the "couldn't care less" > "could care less" shift is why the change occurred. To dismiss the latter form as mere ignorance misses the point that this change is related to systematic difficulties that people have processing negation. That to me is something really interesting about language. So to the extent that this history of this phrase can be tied in to larger issues in language, I'm all for it. When it's used as a prescriptivist club, my reaction is, "meh."
Now before Marina's fans start wielding their own clubs against me, I want to make two points of clarification:
First, I'm not trying to say what people should or shouldn't be interested in. If you like word stories but don't want to explore the systematic parts of language that they suggest, that's your privilege and I don't intend to denigrate that interest. I'm objecting only to a tendency towards false reasoning about language based on these fragmentary stories. That isn't a necessary consequence of anything Marina does in her videos, and I don't think that she intends it.
Second, I'm not claiming that you should use "could care less." If you prefer to use "couldn't care less" for whatever reason, whether it simply feels better to you or whether it gives you a little frisson of pleasure to believe that you are superior to hoi polloi, good on ya, mate. But in this case you should simply use it and not try to dictate your idiosyncratic preferences to others. Neither you, nor I, nor any other individual (even the author of a usage book) is entitled to decide what is correct or incorrect. Linguistic correctness, in so far as that concept has any real meaning, is not determined by a single arbiter. It is negotiated tacitly by the entire community of speakers, and the boat has already sailed on this point: "could care less" is firmly established now and unlikely to go away. Remember that attempts to impose a notion of what is correct from on high have always failed, and will continue to do so. As Samuel Johnson said long ago, "sounds are too volatile and subtile for legal restraints; to enchain syllables, and to lash the wind, are equally the undertakings of pride, unwilling to measure its desires by its strength."
So if you insist on complaining about "could care less" (or whatever your particular grammatical peeve might be) you are only setting yourself up to appear, in a few generations, as ridiculous as Swift now appears for objecting to words like mob or Richard Grant White for objecting to "the house is being built."