I do think the term "logic extremist" sounds rather stupid. Extremism is the antithesis of logic and rationality.
Maybe "logic fundamentalist" or "fundamentalist logician" would make more sense. I think it would have sounded better to use some Vulcan word in the term, such as "T'lahranthi extremist." (I just made that up).
Kor
In non-canon, the term is "Cthia/C'Thia".
And it complicates the discussion, because the term does not translate to logic, but something like "truth-reality". Now anyone familiar with philosophy will see that this has huge implications.
For logic itself is just a tool, as people have said, but Truth-Reality carries a lot of extra meaning - it means that when a Vulcan extols logic, what they are really saying is "realism", "reality", "nature", "naturalism", "truth" - by it's very definition, you could argue being an extremist is the opposite of a naturalistic world-view, since it is founded on speculative 'idealism', rather than the here-and-now. Basically an extremist never adapts their beliefs to fit the facts of the day, or allows for natural change and growth - but a realist by definition is committed to altering their ideals to fit down-to-earth reality.
Also, placing truth as a high-most value, is very interesting, as some people would argue that a society utterly committed to honest self-examination is a healthy democracy, and many of the problems associated with extremism cannot occur in such an environment. Your Nixons, Trumps and My Lai Massacres, and other moral failings, may occur where people neglect the first duty of a citizen - truth. Where it becomes more pressing to rally around your comrades in the army than be honest about having shot 350 civilians. This compounds resentment and at worst, if a society accepts the hypocrisy, it may crumble.