Thank you for these comments. For the embedding into $\mathbf{GL}_n$, one also needs to arrange that the chosen opposite Borel (whose choice is not unique) corresponds to the lower triangular matrices. Can this always be done?

These terms are standard for any reductive group scheme over any base scheme $S$ and were defined in SGA 3, Expose XXVI. By working etale locally on $S$, the question, of course, reduces to the case of a split $G$ (with $S$ still arbitrary).