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 and, except for $\mathrm{E}_8$, he explicitly describes the equations that define the representation. For example, he writes down an explicit homogeneous cubic in 27 variables and states that $\mathrm{E}_6$ is the the subgroup of $\mathrm{GL}(27,\mathbb{C})$ that preserves this cubic form.
For the summary theorem on the linear representations, see Chapitre VIII, Section 10, where he lists each of the lowest representations and notes the various low dimensional exceptional isomorphisms as well.