The authors define a a PRNG as a sequence such that, if you find its distribution (histogram), it resembles the uniform distribution (all values appear the same number of times).
In 1,2,1,2,1,2,..., all values appear the same number of times and its histogram is uniform. However, it is not random. It is clearly not random -- so it is distinguishable from a uniform "true random" sequence, which was my point.
However, as was explained to me by another commenter, what I missed is that you may group a number of consecutive numbers together and find the histogram of that, and, in that case, my sequence fails to have a uniform histogram.
So, at this point I'm willing to accept that the author's definition is correct and I just had a hole in my understanding.