Let $(\Omega,\mathcal{A},P)$ be a probability space and $X_t:\Omega \to \mathbb{R}$ be a stochastic process indexed by, e.g. $t \in [0,\infty)$. Then $X = (X_t)_{t \geq 0}$ is said to have orthogonal increments if $X_t \in L^2(\Omega,\mathcal{A},P)$ and if $$ \mathbb{E}(X_t - X_s)(X_{t'} - X_{s'}) = 0 \quad \text{ for all } \quad [s,t],[s',t'] \subseteq [0,\infty) \text{ such that } [s,t] \cap [s',t'] = \emptyset\,. $$ In the language of Hilbert spaces this just means that $\langle X_t - X_s, X_{t'} - X_{s'}\rangle_{L^2} = 0$ if $[s,t] \cap [s',t'] = \emptyset$.
Now, if $X_t$ is a vector-valued process, i.e. $X_t:\Omega \to \mathbb{R}^m$, then a natural extension of this definition would be just to define an orthogonal increments process as a process such that $X_t \in L^2(\Omega,\mathcal{A},P;\mathbb{R}^m)$ and $$ \langle X_t - X_s, X_{t'} - X_{s'}\rangle_{L^2} = \mathbb{E}(X_t - X_s)^\top (X_{t'} - X_{s'}) = 0 \quad \text{ for all } \quad [s,t],[s',t'] \subseteq [0,\infty) \text{ such that } [s,t] \cap [s',t'] = \emptyset\,. $$ This is, however, not how some textbooks define it. E.g. in P. Caines "Linear stochastic systems" orthogonal increments are defined by (notice the position of the transposed sign!) $$ \mathbb{E}(X_t - X_s) (X_{t'} - X_{s'})^\top = 0 \quad \text{ for all } \quad [s,t],[s',t'] \subseteq [0,\infty) \text{ such that } [s,t] \cap [s',t'] = \emptyset\,. $$ which appears to be a stronger requirement, as $\mathbb{E}(X_t - X_s)^\top (X_{t'} - X_{s'}) = \operatorname{tr}(\mathbb{E}(X_t - X_s) (X_{t'} - X_{s'})^\top)$.
Thus my question:
How are these definitions related and how can one formulate the second definition as a condition on $X_t$ in the Hilbert space $H = L^2(\Omega,\mathcal{A},P;\mathbb{R}^m)$ ?
I would also appreciate any reference, where vector-valued stochastic processes are treated from a functional analytic (e.g. Hilbert space) point of view.