Yes, you are correct. A multi-stage graph is indeed a special type of DAG. However, the difference here is that you cannot have an edge between any two vertices of the same stage. And any edge MUST be from a $S_i$ to $S_{i+1}$, where $S_i$ and $S_{i+1}$ are stages that are topologically sorted. The example below should make it clear:
G=(V, E)
where V = {A, B, C}
and E = {(A,B), (B,C), (A,C)}
The above-given graph is surely a DAG, but it is NOT a multistage graph (because A and C can be put into different stages, but you can't place B in any stage, and moreover you can't create a new stage for B since there is an edge from A to C)
Do let me know if you have any more questions on this :-)
P.S: Usually multi-stage graphs are weighted, but I have neglected the weights here because they are irrelevant here. If you want to be mathematically rigorous, you can assume that all the edges are of unit length.