"No, he is not tall, but short."
Translation:Nem, ő nem magas, hanem alacsony.
Nincs means something is not there, so it is used to negate a noun. But for negating verbs, we use nem.
Think of "nincs" as "nem van". That is what it stands for, basically.
"Magas" is an adjective, and a predicate in itself. It stands after a noun. In that case, we do not use "van". (Hopefully you have seen some of the discussions on this topic.)
So, we can't have a "van", yet, "nincs" would introduce an implicit "van". That is not good. "Nincs" can't be used here. Only the "nem" part of "nem van" is allowed. Which is the word "nem" itself.
With "Nincs jól", the word "jól" is an adverb, not an adjective. With adverbs, we do use "van".
Let's look at the positive versions, to make it clearer:
"Ő magas." - He is tall. "Magas" is an adjective, is the predicate of the sentence, no "van".
"Ő jól van." - He is well. "Jól" is an adverb, it cannot really be a predicate by itself, it needs an added verb. "Jól van" is the predicate (the thing that we are stating).
So, if we negate these two sentences, what happens?
"Ő magas." -> "Ő nem magas." - a very clear case.
"Ő jól van." -> "Ő nem jól van". - some people do say it like this, actually.
But you can also do "Ő nem van jól." - which is not right, you can't say "nem van", you must replace it with "nincs".
So that is how we end up with
"Ő nincs jól." - He is not well.
Technically it could go either way. As complicated as Hungarian is, the third person pronoun does not distinguish. Some words in Hungarian do use nő endings to indicate that the noun is female, such as barátnő, which is a word you could use to mean "girlfriend".
Fun fact, every occurrence of magas in this lesson is wrong. Mag means "seed", and magas would mean "seedy". Magos is the word you're looking for here.