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.


**Added remark:** Cartan continues to refer to groups of type $B$ and $D$ merely as "the largest groups preserving a quadratic form in $n$ variables" or similar terms for a long time.  Even in his papers of 1913–1915 classifying the real forms of the complex simple Lie groups, he uses such terminology, though he clearly finds the special case of the compact real forms of special interest.  

The first place that Cartan actually refers to 'orthogonal groups' that I can recall are in his 1926–27 papers on the classification of Riemannian symmetric spaces.  There, he begins referring to any subgroup of $\mathrm{GL}(n,\mathbb{R})$ that preserves a quadratic form as '*an* orthogonal group' and then, later, finally calls the maximal group that preserves a positive definite quadratic form '*the* orthogonal group'.  I don't recall when or whether he used any notation such as $\mathrm{O}(n)$ or $\mathrm{SO}(n)$.

Whether the term 'orthogonal group' was original to him, I can't say.  By that time, of course, Weyl had already started his research on compact Lie groups, and it may be that Weyl had already used the term 'orthogonal group' well before Cartan.