Teaching changes in "demand" v. "quantity demanded"
I think the terminology below is pretty unambiguous and also common: 



  a shift in the demand curve 



to refer to movements from $D$ to $\tilde D$ (using shift is quite more specific than using change, as it more clearly refer to a change in the whole demand schedule).



  a movement along the demand curve



to refer to changes in $D(P)$ induced by movements in the supply curve (this I normally do it by showing the "dynamics" in the graph, from the original equilibrium to the new equilibrium along the demand curve). 



  a change in the quantity demanded



to compare different $D(P)$ in the horizontal axis. A mention to the concept of equilibrium can be useful here too. For example, a change in the final/equilibrium quantity. Here the demanded adjective is in a sense redundant, as in the equilibrium the quantity demanded and supplied is the same, but if you are being very forma as in $D(P)=S(P)$ then its fine. This terminology is different from the first one (shift in the demand curve) in the sense that the latter is a ceteris paribus exercise, whereas the former is not (unless supply is perfectly in/elastic).



Regarding your question about whether to use the textbook's proposed terminology, I would not advice so. First of all, it is not clear what the authors propose to use to refer to changes in $D(P)$ induced by "changes in the demand [curve]". Why to have a separate notation for such changes and those induced by changes in the supply? That is very arbitrary. 

Additionally, as their restrictive terminology is probably not the one you are used to (judging by question and comments), forcing you to adopt such incomplete terminology (which is likely not to come to you naturally) might confuse your students because you might not always consistently use it. Provided you want to keep using that textbook, I would tell them something like 


  beware of this very restrictive termonology issue, which is uncommon in other books. More common instead is this terminology, and what I normally use is this. In any case, all this is not very relevant for understanding. In your answers, always explain what you mean, ideally with the aid of annotated graphs. Correct answers (mostly) do not depend on terminology.