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.

ADDED: To be more precise (while keeping most of the heavy notation out of the way), note that all three connected reductive groups given in the statement of the problem share a common semisimple derived group $H$ (up to isomorphism) which has a finite center $Z(H)$.    Now embed the centers (both tori) of the two target groups in a large enough torus $S$ defined over $\mathbb{F}_q$ and use the Cabanes-Enguehard construction to get an ultimate target group $S \times H / Z(H)$; here $Z(H)$ acts diagonally.