Question: Recently I’ve got stumbled upon the question: What’s the difference between *quite a lot*, *quite a few*, and *quite a bit*?

This is very confusing considering *a lot* and *a few* have almost opposite meanings.

As far as I can see in dictionaries *quite a <COUNT-WORD>* is always *many*. Is that so, or am I missing some important difference?

Answer: 
Regarding the fact that I agree with you about the same meaning of *quite a <COUNT-WORD>* and *many*. I think these there have the exact same meaning but they better be used when they fit in the context of the sentence although they can be replace each other pretty freely. For example, I

* It's better to say *"He wrote quite a few words about recent incident in UK lately."* than using "quite a bit of words".
* Or *"He has quite a lot of knowledge in this field"* rather than using *quite a few knowledge* or better to rephrase your sentence if you like to use it like: *"He knows quite a few facts about this subject"*.
* Or *"There is quite a bit of information in this book listed about the theory of Compute Science"* is make more sense than using the others.

So, the only thing that I can emphasize is to use ***a few*** with countable word and try to pick the best that fits to the context of the sentence.
Also, I would probably say "There are *quite a lot* of words on the page", but "There are *quite a few* words on the page" also sounds correct. However, "quite a bit" doesn't work in the same context.