I fed rye to a very inactive white starter and it literally skyrocketed overnight.

I had the exact same experience, but with a little whole wheat flour rather than rye, and it took my 'stagnant' starter into over-drive overnight.  Thanks to ehanner for that advice.  He said that plain AP flour is good enough for nutrients most of the time, but that it's good to give it a shot of WW once in awhile.  Nice trick to remember, and it looks like rye works the same way.  Not sure why it works, but I'm guessing there's a food v. nutrient v. metabolism thing going on.

Your rye additions will make the sourdough starter, and resulting breads, more sour.  Now that you've boosted your starter, you can go back to straight white AP flour feeding if you like or if you don't prefer the extra sourness from the rye.

I wonder if a prudent feed with WW (or rye) flour while trying to start a new starter might be a good way to boost it and hurry things along?   Hmmm....

Brian

So, when making bread with this using white flour, will it taste like white sourdough?

Even better.  Gives a depth to the sour flavour.   To quote Mariane its like putting the starter on steroids.  Naturally of course.

Semolina and spelt flour have a similar effect as rye in terms of an immediate boost in activity. The yeast cells are simply reacting to the higher percentage of available nutrients found in these flours that aren't present in typical white bread flours.

Franko