Thanks for the nice problem. First, let us scale the variables
$f \rightarrow x$, $\zeta \rightarrow \sqrt{\Gamma} \, \xi$,
and consider the Langevin equation
$$
\dot x = -k \, x + \sqrt{\Gamma} \, \xi,
\qquad (1)
$$
where $\xi$ is a colored noise satisfying
$\langle \xi(t) \, \xi(t') \rangle = e^{-\gamma \, |t-t'|}$.

We wish to show that the Fokker-Planck equation for $P(x, t)$ is given by
$$
\frac{ \partial P } { \partial t } = \frac{ \partial } { \partial x } ( k \, x P)
+
\frac{ 1 - e^{-(k+\gamma) t} } {k + \gamma} \Gamma
\frac{ \partial^2 P } { \partial x^2 }.
\qquad (2)
$$

## Solution is a Gaussian

The idea van Kampen (page 243) is to treat $\xi$ as a Ornstein-Uhlenbeck process
(i.e., the position of a harmonic oscillator under
a white-noise Brownian dynamics).
Then, the extended equations of motion are
$$
\begin{aligned}
\dot x &= -k \, x + \sqrt{\Gamma} \, \xi, \\
\dot \xi &= -\gamma \, \xi + \sqrt{2 \gamma} \, d W(t),
\end{aligned}
\qquad (3)
$$
where $W(t)$ is the Wiener process (so $dW(t)$ is a white noise).

The distribution $P(x, \xi, t)$ for Eq. (3) satisfies
$$
\frac{ \partial P(x, \xi, t) } { \partial t }
=
\frac{ \partial } { \partial x }
\left[
\left(k \, x - \sqrt{\Gamma} \, \xi \right) \, P(x, \xi, t)
\right]
+
\frac{ \partial } { \partial \xi }
\left[
(\gamma \, \xi ) \, P(x, \xi, t)
\right]
+
\gamma
\frac{ \partial^2 } { \partial \xi^2} P(x, \xi, t).
$$
Since this is a linear Fokker-Planck equation,
the solution is a joint Gaussian distribution
for $x$ and $\xi$.
Further, the deterministic part for $x$
is unaffected by the colored noise $\xi$.
So the marginal distribution of $P(x, t)$
must adopt the form of
$$
\frac{ \partial P(x, t) } { \partial t }
=
\frac{ \partial } { \partial x }
\left[
k \, x \, P(x, t)
\right]
+
B(t)
\frac{ \partial^2 } { \partial x^2} P(x, t),
\qquad (4)
$$
where $B(t)$ is a function to be determined
(which would be a constant if the noise were white).

## Determining $B(t)$

By multiplying Eq. (4) with $x^2$ and integrating,
we get
$$
\begin{aligned}
\frac{ \partial \langle x^2 \rangle } { \partial t }
&=
- 2 \, k \, \langle x^2 \rangle
+ 2 \, B(t).
\qquad (5)
\end{aligned}
$$

On the other hand,
the explicit solution of Eq. (1) is
(assuming $x(t = 0) = 0$ for simplicity),
$$
x = \sqrt{\Gamma} \int_0^t \xi(\tau) \, e^{-k(t-\tau)} \, d\tau.
\qquad (6)
$$
By multiplying Eq. (1) by $2 \, x$, and averaging, we get
$$
\begin{aligned}
\frac{ d \langle x^2 \rangle }{ dt }
&=
- 2 \, k \langle x^2 \rangle + 2 \sqrt{\Gamma} \langle x(t) \, \xi(t) \rangle
\\
&=
- 2 \, k \langle x^2 \rangle + 2 \, \Gamma
\int_0^t \langle \xi(\tau) \xi(t) \rangle \, e^{-k(t-\tau)} \, d\tau
\\
&=
- 2 \, k \langle x^2 \rangle + 2 \, \Gamma
\int_0^t e^{-\gamma (t-\tau)} \, e^{-k(t-\tau)} \, d\tau
\\
&=
- 2 \, k \langle x^2 \rangle + 2 \, \frac{ \Gamma } { \gamma + k } \left( 1 - e^{-(\gamma + k) \,t} \right).
\end{aligned}
$$
where we have used (5) on the second line.
Comparing this to (5), we get
$$
B(t) = \frac{ \Gamma } { \gamma + k } \left( 1 - e^{-(\gamma + k) \,t} \right).
$$
So the Fokker-Planck equation for $P(x, t)$ is given by Eq. (2).

## Discussion

This example is instructive, because with a non-white noise, the magnitude $B(t)$ depends on the noise itself, but also the deterministic force (through $k$).

Both are relevant, and "the misconception that Langevin equation is the universal stochastic differential equation for all kinds of noisy systems is responsible for the difficulties mentioned"* in your post.

Take the SDE from Thomas' answer,
$$\frac{dy}{dt} = A(y) + C(y)L(t)$$
where $L(t)$ is the noise term. Suppose we can turn the noise off, so we'd only be looking at the isolated, deterministic, system, and suppose that we got:
$$\frac{dy}{dt} = A(y)$$
It is thus obvious that the noise term in this case must be interpreted as per Stratonovich, because an Itô interpretation would change the dynamics of the isolated system.

Above we had assumed, as one does, that $\langle L(t)L(t') \rangle = D \delta(t-t')$, but rarely are actual physical processes made out of Dirac deltas. Now if they are not, then by Wong-Zakai, Stratonovich is the only possible interpretation here (i.e. if $L(t)$ is more general, and as it approaches a delta, Stratonovich is the integration form that arises).

Now by turning the noise on and off we have arrived at the conclusion that Stratonovich is the only way forward, but in the very beginning I said that both formulations are relevant. Indeed, what if the noise cannot be separated out?

The important distinction is to make between external and internal sources of noise. We dealt with the external, but what if the noise is internal, and is impossible to turn off like say in chemical reactions? It's not that we have a deterministic process with a noise term slapped on top, we have a stochastic process and one might be able to argue that the averages behave in a deterministic manner, but by no means can then one then simply just add a noise term back on top without more justification: "For internal noise one cannot just postulate a nonlinear Langevin equation or a Fokker-Planck equation and then hope to determine its coefficients from macroscopic data. The more fundamental approach of [the expansion of the master equation] is indispensable."

* Any quotes in my answer have been taken from the definitive book on the matter: *Stochastic Processes in Physics and Chemistry* by van Kampen, which I strongly suggest you consult for more detailed explanations of the issues and how one goes about dealing with them.

## Best Answer

This question is a little too big, entire text books have been written to answer it. A standard reference is van Kampen, Stochastic processes in physics and chemistry.

Roughly speaking, for a Markov process

Master equation -> Kramers-Moyal expansion -> Fokker-Planck equation

where the master equation gives the microscopic probabilistic rule for transitions in some configuration space, and the Fokker-Planck equation is the corresponding equation for single particle probabilities.

The Langevin equation is a simple stochastic model equation designed to give the same FP equation as the master equation. The Boltzmann equuation lives in a larger space (phase space), and is not stochastic, but is again designed so that linearized Boltzmann is equivalent to FP.

[This make it sound as if the Boltzmann equation is just some kind of model equation. This is not correct. There is a separate path to the Boltzmann equation, starting from the classical Liouville or quantum von-Neumann or quantum Kadanoff-Baym equation.]