Teaching changes in "demand" v. "quantity demanded" I am always struggling with teaching this one. 

The terminology of standard intro textbooks is unfortunate, although it's so pervasive that we probably have to live with it.


By a "Demand", intro textbooks really mean a "Demand function" (but talk about "Demand" either for concision, or because they think the word "function" will scare intro students).
Having shied away from calling a function a function, intro textbooks then cannot speak of "the demand at price $P$" (just like you would speak of the value of $f(\cdot)$ at $x$), and need to introduce the awkward terminology "quantity demanded at price $p$".


That's an ok compromise, until you reach the usual section on "changes in the quantity demanded v. changes in demand".

There, students are often told that we should only talk of a "change in the quantity demanded" if the demand function $D$ is fixed, and that such "changes in the quantity demanded" must be the consequence of a change in price only (typically following a change in supply).

Then how can I speak of the comparison between $D(P) &gt; \tilde{D}(\tilde{P})$ without being terribly confusing to my students? 

If I say, 


  1) "Following a change in demand from $D$ to $\tilde{D}$, the equilibrium price changed from
  $P$ to $\tilde{P}$ and the quantity demanded therefore changed from $D(P)$ to
  $\tilde{D}(\tilde{P})$",


I think I am guaranteed to confuse the hell out of them, because the textbook essentially tells them that "we should never talk of a 'change in quantity demanded' unless the demand function remains unchanged".

At the same time, is there any other ways to formulate 1)? 


Should I use a different word and speak, e.g., of a "move" or a "movement" in quantity demanded from $D(P)$ to $\tilde{D}(\tilde{P})$  to avoid any confusion with a "change in quantity demanded".
Or is it better to be explicit about the ambiguity?
Or is it better to try and avoid the issue as much as possible?


What is your experience teaching this? What are your tricks to get around this terminological difficulty? (Or, maybe, why do you think it is not a terminological difficulty at all?)
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.