Half-wave diode rectifier - is textbook wrong?
Yes, you are right, have a look at this ltspice simulation of a simple full wave rectifier (click to enlarge):



Textbooks like to simplify things before they go in depth (if at all). How many text books have you seen to talk about diode drop at that point at all? 

Its an application of wittgensteins ladder.

Note that at higher frequencies things like the diodes recovery time will start to play an important role too, but even less textbooks talk about that. Both things are not immediately important to understand the concept that should be learned at that point.