I don't want to offend anyone.
It is probably a sign of today's times that subjects related to mathematics, programming and algorithms are expressed by people who have no clue about it. They imagine how it should work and believe that they are right even without reading even one scientific paper on the subject.
Yep. Hence there is always latency when applying digital pitch shifting. However, I stand by my assertion that input pitch recognition is not needed for a chromatic pitch shifter. Only an envelope detector is required which requires less samples of the digitized waveform then a pitch detector.
I speculate that the harmonic pitch shifter, on the other hand, requires both an envelope detector and a frequency detector. Both pitch shifters will then use some kind of algorithm similar to the one you linked.
Edit... After considering this a little more, I'm definitely minimizing the complexity of the process. For any pitch shifting, some sort of frequency analysis most likely takes place to provide a realistic sounding shift. I've talked myself out of this argument (don't tell my wife!)