That's a reasonably accurate description. It's aimed at an unsophisticated audience ( i.e. little scientific background) so he is using some simplifications and easily understood explanations for a very complex process.
Yes water injection increases the effective octane.
Yes one of the ways it accomplishes that is by cooling the peak combustion temperatures.
No it does not significantly modify the fast burn portion of the combustion process, but its effects on initial cylinder temps and pressures, tends to slightly modify the early stage of combustion, and slows things down a bit. The end result is a slight delay in reaching peak cylinder pressure.
The ideal point for peak cylinder pressures varies some from engine to engine, but he is correct that excess early pressure rise before the piston is past about 11 deg ATDC is usually a contributing factor to detonation damage, as the piston is not able to quickly accelerate and relieve high pressures and capture the energy as useful work. If the peak pressure comes too soon the mechanical arrangement of the connecting rod and the crank shaft is very nearly in line, and most of the pressure is transferred directly to high loading in the crank and con rod bearings, and side thrust on the piston which causes high friction.
Short answer the book is accurate enough for a general understanding of the processes, with out being confusing. There is lots of highly technical info out there like the paper at :
http://www.waterinjection.info/
and the papers published by NACA and others that go into very great detail about how WI works.
Larry