What I'm referring to is the "now-cast", but his other two definitions both seem to shy away from saying "this is flat-out the probability we think of the election".

The point is, you can redefine or choose a definition of probability if you want, but if it's less useful than the normal definition (and confusing to people!) then people are free to criticize your work on that basis.

And there's a very useful, testable, mathematical definition of probability that allows us to equally assess everyone's predicting ability, and Nate Silver is dodging it.

If you're interested in this subject, there's a non-mathematical discussion somewhere in Tetlock's book Superforecasting which is interesting in general.