Well, I thought the same. That's what I was taught as a kid. But I looked it up and it seems "can not" is permissible too, though "cannot" is definitely preferable. "Can not" should be accepted, though if it is accepted and just marked as a typo I don't think it's a big deal.
Hmm, I think that conclusion is largely to misread the sources you provide. They ok things like "can not only ride bicycles but also unicycles," which is a different use, not about inability/hindrance at all (usually probably quite the opposite). One says it's ok for emphasis. That makes sense, too, inasmuch as one might well pronounce it "You can [pause] not [pause] do that" to really hit the point home, but that really probably has more to do with the always flexible matter of representing very oral patterns in writing.
Can Not is permissible in both written and spoken English. "He can not read" simply is more emphatic than "He can't read". For example, you could say "Don't you understand? He can't read!" or "He can not read." The meaning and emphasis is a little different but both are 100% grammatically correct.
Plagiarised from szeraja_zhaba here: https://www.duolingo.com/comment/11901887
Уме́ть means having a certain skill.
Я уме́ю гото́вить. 'I can cook.' = 'I know how to cook.' Я уме́ю пла́вать. 'I can swim.' Я уме́ю игра́ть в футбо́л. 'I know how to play f̶o̶o̶t̶b̶a̶l̶l̶ soccer.' Я уме́ю чита́ть мы́сли. 'I can read [other people's] mind.' Мо́чь refers to general possibility, often the possibility to do action at the moment. When used without any context, it usually sounds as an offer:
Я могу́ гото́вить. 'I can cook', 'I could cook'. We don't say things like «я могу готовить» often, because, well, who doesn't? I think it's used with «не» much more often than in positive sentences.
Я не могу́ гото́вить, не́ из чего. 'I can't cook, I don't have anything to cook!' Я не могу́ пла́вать без пла́вок. 'I can't swim without swimming trunks.' Я не могу́ игра́ть в футбо́л без мяча́. 'I can't play soccer without a ball.' Я не могу́ чита́ть его мысли, пока́ он но́сит ша́почку из фольги́. 'I can't read his mind while he wears a tinfoil hat.'