You can look at your process $X_{t}$ as a two dimensional stochastic process
$$Y_{t}=\left[\begin{array}{cc}X_{t}\\ \eta_{t}\end{array}\right]$$
Then
$$dY_{t}=\left[\begin{array}{cc}dX_{t}\\ d\eta_{t}\end{array}\right]=\left[\begin{array}{cc}b(X_t)+\lambda\eta_{t}\sigma(X_{t})\\ \lambda\eta_{t}\end{array}\right]dt+\left[\begin{array}{cc}\alpha\sigma(X_t)&0\\ 0&\alpha\end{array}\right]\left[\begin{array}{cc}dW_{t}\\ dW_{t}\end{array}\right]$$
and the infinitesimal generator is of the form
$$LV(y)=LV(x,\eta)=\left(b(x)+\lambda\eta \sigma(x)\right)V'_{x}(x,\eta)+\lambda\eta V'_{\eta}(\eta,x)$$
$$+\frac{1}{2}\alpha^{2}\sigma^{2}(x)V''_{xx}(x,\eta)+\alpha^{2}\sigma(x)V''_{x\eta}(x,\eta)+\frac{1}{2}\alpha^{2}V''_{\eta\eta}(x,\eta)$$
By the way, the infinitesimal generator of an Ornstein-Uhlenbeck process of the form
$$d\eta_{t} = \lambda\eta_{t} dt + \alpha dW_{t}$$
is
$$LV(\eta)=\lambda \eta V'(\eta) + \frac{\alpha^2}{2}V''(\eta)$$
Starting from
$$dY_1=-\frac12 Y_1 dt-Y_2 dB_t, \\
dY_2=-\frac12 Y_2 dt+Y_1 dB_t, $$
and using the general formula for the generator of a diffusion process $dX_t=f(X_t)dt+g(X_t)dB_t$, which reads
$$\mathcal{A}=\sum_{i=1}^n f_i(x)\partial_{x_i}+\frac{1}{2}\sum_{i=1}^n\sum_{j=1}^m g_{ij}(x)\partial_{x_i}\partial_{x_j},$$
we have with $n=2$ and $m=1$
$$\mathcal{A}=\frac12\left(-y_1\partial_{y_1}-y_2\partial_{y_2}+y_2^2\partial_{y_1}^2+y_1^2\partial_{y_2}^2\right)=\frac12\partial_{\theta}^ 2.$$
Which is one half times the Laplacian on $S_1$.
In fact, one of the definitions of Brownian motion on a Riemannian manifold $(M,g)$ is that its generator is $\frac12\Delta_g$, where $\Delta_g$ is the Laplacian of $g$. Thus in order to find the generator of Brownian motion on $S_2$ in Cartesian coordinates, take the Laplacian on $S_2$ and transform it to Cartesian coordinates (a slightly tedious calculation).
Best Answer
To be clear even in $\mathbb{R}^{n}$, one again needs to start from some generator and then define the corresponding semigroup (satisfying the Cauchy-problem) and stochastic process. For Brownian motion we picked that particular transition density precisely because it satisfies the heat equation.
Once some basic processes were built, then this procedure was back-engineered too eg. see Feynman-Kac, of the correspondence, so one can start from an SDE and figure out the corresponding generator.
For the math picture, take a look at Examples 3.3.2./3.3.3. in "Stochastic Analysis On Manifolds" by Elton P. Hsu. He derives the generator for Brownian motion on the sphere.
For a possible physics-picture, see "Brownian self-driven particles on the surface of a sphere" or "A note on the exact simulation of spherical Brownian motion". For example, one cool application is bacteria on the surface of cells