Leaving aside your minor changes in the notation and definition of Cabanes-Enguehard (who also adopt some arbitrary notation), my understanding of their exercise is that it responds to the obvious non-uniqueness in the target group. Of course, the whole problem originates in embeddings like the one of a special linear group in the corresponding general linear group when the former group has a nontrivial (but finite) center.
The target group always has a connected center, which is just a torus of some dimension (defined over $\mathbb{F}_q$, though this doesn't seem to enter directly into the question here). So the basic concern is that two different target groups may involve central tori of different dimensions. I guess the natural solution is to embed both of these tori into a possibly larger one (still defined over $\mathbb{F}_q$), which in turn allows one to construct a common target group for both of these as in the standard construction used by Cabanes-Enguehard. However you approach this you run into some arbitrary choice of central torus, so there is no universal construction. Am I oversimplifying the question? I got lost in the next-to-last paragraph of your discussion.