Understanding the Diameter Constrained MST Problem
The diameter of a graph is the longest shortest-path distance over all pairs of vertices. This distance can be measured in terms of edge weights or in terms of number of edges (which is known as geodesic distance). The measure that you choose depends on your particular application. In the paper that you mention, the authors focus on geodesic distance.

When you want to obtain a Minimum Spanning Tree (MST), you focus on edge weights because all spanning trees have the same number of edges: $|V|-1$.

Aiming for an MST that is constrained by a maximum diameter is perfectly possible. The authors of the paper are not setting two conflicting optimization goals, but, rather, just adding an additional constraint to the MST problem. Adding this contraint, however, turns the MST problem into an NP-Hard problem (even though computing the diameter of a graph and obtaining an MST, separately, are problems that can be solved in polynomial time).

To gain further insight into the notion of diameter (in the context of MSTs), consider the following example:



The spanning tree on the left (whose edges are highlighted in red) is minimum. Its total weight (i.e., the sum of the weights of its edges) is 7 and its diameter is 5 (because the vertices that are farthest from each other are 5 edges apart). If we add the constraint that the diameter must be at most 2 (that is, $d=2)$, then the diameter-constrained MST is the tree on the right. Its total cost is 12 (higher than the tree on the left), but it is the only spanning tree whose diameter (ignoring weights) is $\leq 2$.