Question: I was just reading an e-book and saw this sentence which really confused me:

[![enter image description here](https://i.stack.imgur.com/CuYCV.png)](https://i.stack.imgur.com/CuYCV.png)

"Played twice as fast as written" - This isn't what cut time actually means right, or are there situations where cut time means exactly that?

I thought cut time is basically just 2/2 time signature, but the speed would still depend on the bpm and the note durations...

Answer: 
No, it's not current... and it doesn't really make much sense. (How fast is it "written"?)

Your assertion that time signatures do not dictate tempo is correct. Certain meters might imply faster tempi (6/4 is probably going to be used for slower pieces, and 12/16 is usually seen in fast pieces like gigues) but those are general usages, not requirements.
The e-book was trying to contrast two different meters, 2/2 to 4/4, kind of assuming that the reader might not notice or understand the difference at all. They were trying to provide a simplified first explanation of an unfamiliar meter symbol.

They meant "Crossing the C has the effect that the conductor's hand will start moving in cycles of two beats rather than four beats and you players will have to play twice faster to keep up." 

Except that there's no conductor around which makes the entire difference more abstract to formulate and less constrained to execute.