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: 
It may be that the e-book used the same notation example written in 4/4 earlier, and is indicating to play this version faster? 

You are correct that the time signature is not the indicator for tempo. There is an old tradition of using Alla Breve to indicate the piece is a faster tempo, but current practice is to use tempo markings. The cut time choice changes the feel of the music because of the strong beat.
In a simple sense, it is correct. "Whatever tempo you have in mind, this is twice as fast as that". This works on the assumption that most people are used to the quarter note being the beat, which happens to be true.

But it basically means that the pulse of the song will be on the half-notes. If someone was standing there waving a baton in front of you, they would be counting the half-notes, and counting two beats per measure (2/2 time as you said).

If it says "moderato" on top of the page it would refer to the speed of the half-notes and if there is a bpm marking it would also refer to the half-notes, although it's common to make this explicit by writing "half-note = tempo" (see top two examples here <http://www.janvanbiezen.nl/bachfig01.gif>).

This is common in marching band music where writing in 2/4 would look awkward (I guess?).