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

"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...

It would be more accurate to say that cut time "will sound twice as fast as the same notes played in 4/4 at the same tempo". That's essentially what they're trying to get across.

But even that wouldn't really be accurate. Cut time is a duple meter, 4/4 is a quadruple meter. The difference is subtle, but it's still a difference.

That sentence "Played twice as fast as written" indicates that someone must have a misunderstanding. Someone who probably thinks that quarter notes are supposed to be played at a certain speed. That person would need more knowledge and experience with both tempo markings and different kinds of time signatures.

I suppose you could say that in the beginning when you learn your first note values and make your first exercises with the values you do get used to think of quarter notes as indicating some basic speed. You need to crawl before you can walk so to speak. But you certainly better learn to walk before you write a work book on the matter.

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.

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.

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?).

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.