Your quote about Cartan thinking of $B_n$ and $D_n$ as 'projective groups..." is actually Cartan describing the lowest dimensional homogeneous space of these groups (except, of course, for a few exceptional cases such as $D_2$, which is not simple, and therefore should be left out of the description).
If you go just a little bit further in Cartan's 1894 Thesis, to Chapitre VIII, Section 9, you'll see that Cartan describes linear representations as well. For example, of $B_\ell$, he says "C'est le plus grand groupe linéare et homogéne de l'espace à $2\ell{+}1$ dimensions qui laisse invariante la forme quadratique $$ {x_0}^2 + 2x_1x_{1'} +2x_2x_{2'} + \cdots + 2x_\ell x_{\ell'}" $$ with a similar description for $D_\ell$.
In fact, he gives the lowest dimensional representation of each of the simple groups over $\mathbb{C}$, including the exceptional ones.