"...these special elections where, by definition, very few people turn out to vote..."
Really? That's the definition of a special election? That very few people turn out to vote?
I thought the definition of a special election was an election to fill a political office that has become vacant between regularly scheduled elections. (Well, in truth, I have just checked on Wikipedia in order not to make an idiot of myself, because, although my knowledge of American politics is significantly better than it was in my pre-West Wing days, it's still far from complete.)
Anyway, back to the point at hand. The definition of a special election is not one in which few people turn out to vote. Therefore, the phrase "by definition" is not correct here. It would work in a phrase like this one:
"... these special elections, which by definition are not held in November..."
or even "... these special elections, which by definition are usually unexpected..." That is stretching it somewhat, but we can surmise that if they are not "regularly scheduled" it's because something has happened out of the blue. It's shaky ground, but you could just about argue it still stems from definition.
But low turnout? Of course, that's part of the nature of special elections. (Or by-elections, as we Brits call them.) So the phrase they wanted was by their very nature, not by definition.
Why does this matter? I was going to say, it doesn't really, so you can all stop listening to me now. And you can, but it does: precision in language is important. For accurate communication. For mutual understanding. And also, most importantly of all, just because.