I use a definition of normal quasi-uniform spaces from this article.
Now I have proved (I do not present the proof because it uses "funcoids" which can be read about only in my manuscripts.) that every uniform space is normal. Is my proof with an error? (I think so, because nLab says "it is not generally true that uniform spaces are normal (so that separated ones would be $T_4$), because for instance an uncountable power of the real line (with its usual topology) is not a normal space."
So is my proof in error or the term "normal" in the above mentioned arXiv article and the above mentioned nLab quote mean two different things?
I cannot evaluate your proof, but a "normal quasi-uniformity" is definitely not the same as the topology from the uniformity being normal. It's true in general that terms like "normal", "regular" etc. are overused. There is also a notion of normal covers, normal family etc., all of which are unrelated.
The linked-to article shows that every topology of a completely regular space is generated by some "normal quasi-uniformity", and there are many completely regular and not normal topological spaces (like the uncountable power of the real line, or the Sorgenfrey plane, the Niemytzki plane, etc.), which already shows this. So if you reprove this fact (using their definition of normal quasi-uniformities, or something provably equivalent to it), you're probably fine.