Nem tudom is the so-called definite conjugation. There is an object implied whose properties are known. It translates to "I do not know that." (You can also say Azt nem tudom to spell out that object.)
Nem tudok is the indefinite conjugation and there's no determined object. You can use that conjugation for saying that you can't [verb]. Like nem tudok írni - I cannot write.
If you write "I can't dance", you're also not translating, because tudni is "know" and not "can".
But "I don't know to dance" is not English.
If you say that tudni can also be "can" because that's how you would rather say it in English, then I think it's reasonable to also accept "know how to" because that is also a way to say it in English.
I think it's reasonably to consider "can, be able, know how to" synonyms.
"Tudni" is actually the direct translation of both "to know" and "to be able to". In my opinion. The reason for inability can be various, just like in English.
As far as "can", there are some differences in usage. I would dare to say, you can translate "can" with "tud" wherever "can" can be replaced with "be able to".
Tud is one of the best translations you get for the concept of 'can', though. It can mean either being physically able to doing something, or knowing how to do something. So I'm sure both ways of expressing that should be accepted.
It's not really paraphrasing cause the English language doesn't use the same grammar when it comes to tud.
I don't think it's reasonable to treat "tud" as "know" exclusively, though. It sounds prejudicial to force a Hungarian verb into one English concept. "Tud" isn't about cognitively acquirable skills quite often. In a situation where knowing how to do something doesn't imply you can do it yourself - "tud" will hands down refer to the latter.
To mention an interesting experiment: a bicycle with inverted steering is terribly difficult to actually ride while everyone can easily understand how it should be ridden. In this case, the fair thing to say would be "nem tudok biciklizni a fordított kormányzású biciklivel", even if you know how to ride it in theory.
And Spanish 'saber' and 'conocer' as well as Slovak 'vedieť' and 'poznať', Esperanto 'scii' and 'koni' or Czech 'vědět' and 'znát'. Should I continue?
By the way I am really suspicious that the Esperanto infinitive verb ending -i actually comes from Hungarian.