Other books, written by physicist that actually now for sure about the properties of light will all say that an observer in motion will have two light beams traveling the same distance reach him at the same time as according to the M&M experiment.
I had a friend like you thought the same thing, I got him interested in it and we would talk about different things in theoretical physics. He then got the same book you did, and then I told him that it wasn't exactly right. That is only because I had already read other books about it that say something completely different. I have read books on just light itself, nothing else. It is a strange thing and an interesting topic if you know the right people.
But, sometimes straight from the horses mouth isn't the best way to go, and then sometimes it can be misleading. Einstein was a smart guy, but he wasn't right about everything all the way through. Just get a book about light, and see what it says. It will not agree with the thought experiment. Experiments have been done that test this. The only thing that is found to change from on object in motion is the wavelength of light. Not the speed or arrival times.
I know it could be hard to believe, but I speak the truth. Einstein use to always say, "God does not play dice with the universe!". But then they still came up with quantum theory and the uncertainty principle, he was actually completely wrong about quantum mechanics. They went and discovered it against what he thought anyways. He really didn't discover much after this point.