Gas evolution in reaction between zinc and HCl
When HCl reacts with zinc or any other active metal such as aluminium, hydrogen is produced. This is what your textbook tells you. Whats is perplexing is the appearance of "fog" in the Erlenmeyer. In the beginning there is air above the liquid. When hydrogen evolution starts, the flask begins to fill with $\ce{H2}$ gas and creates enough pressure to displace the column of water in the buret. So that part of your observation is correct and you see a colorless gas displacing water column in the buret. What is in the conical flask that is causing the fog? If you were to touch the Erlenmeyer you could have noticed that it was hot, secondly there was a very vigorous evolution of hydrogen bubbles. There are several reasons, each contributing to the fogginess in the flask. That "fog" is just not hydrogen, air but vapors and very fine droplets of acid.

(i) creation of very fine droplets of acid by vigorous bubbling process (major contribution)
(ii) steam generated by the heat of the reaction
(iii) Hydrochloric acid has the tendency to produce fumes in moist air (seen with very concentrated acid). You did not mention the concentration of the acid used.