The generalization of Newton's 2nd law to general relativity is given by
$$ m \frac{d^2 x^\mu}{d\tau^2} + m \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = f^\mu \qquad (\star)$$
where $\tau$ is the proper time along the particle's worldline, $f$ is the net 4-force acting on the particle, and $\Gamma^\mu_{\alpha\beta}$ are the Christoffel symbols corresponding to your choice of coordinates.
In inertial Cartesian coordinates, all of the $\Gamma$s are equal to zero which means that
$$m \frac{d^2x^\mu}{d\tau^2} = f^\mu$$
If you pick different coordinates in which the $\Gamma$s are not zero, then obviously you're going to need to include that extra term. This happens if you use polar coordinates, for example, but it also occurs when you use accelerated Cartesian coordinates. In the latter case, what you can do is simply move the extra terms to the other side of the equation and call them pseudoforces:
$$m \frac{d^2 x^\mu}{d\tau^2} = f^\mu - \underbrace{m \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau}}_{\text{pseudoforce}}$$
Example: The coordinates which correspond to a relativistic observer undergoing constant proper acceleration are the Rindler coordinates. In this coordinate system, assuming proper acceleration $a_0$ along the $x$-axis, the line element becomes
$$ds^2 = -\left(1+ \frac{a_0 x}{c^2}\right)^2 c^2dt^2 + dx^2+dy^2+dz^2$$
The nonzero Christoffel symbols are
$$\Gamma^0_{10} = \Gamma^0_{01} = \frac{a_0/c^2}{1+a_0 x/c^2}\qquad \Gamma^1_{00} = \left(1+\frac{a_0 x}{c^2}\right)a_0/c^2$$
which means that the relativistic generalization of Newton's 2nd law for $x$ is
$$ m \left(\frac{d^2x}{d\tau^2} + a_0 \frac{1+\frac{a_0 x}{c^2}}{\left(1+ \frac{a_0 x}{c^2}\right)^2 - \frac{v^2}{c^2}}\right) = f_x$$
In the nonrelativistic limit, this becomes
$$m \left(\frac{d^2 x}{dt^2} + a_0\right) = f_x \iff m \frac{dx^2}{dt^2} = f_x \underbrace{ -\ m a_0}_{\text{pseudoforce}}$$
which is exactly what we'd need to do in Newtonian mechanics if we wanted to switch to an non-inertial frame accelerating with $\mathbf a = a_0 \hat x$.
So in summary, you have the following options:
- Choose a coordinate system in which all the $\Gamma$s vanish - that is, a global inertial frame. Such frames generally do not exist in curved spacetime, so you can only do this in the context of SR.
- Recognize and accept that the left-hand side of $(\star)$ has additional terms due to the $\Gamma$s, which reflect the fact that your coordinate basis changes with position. If you've ever constructed Newton's laws in polar coordinates, this is what you do.
- Move the term $m\Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau}$ to the right-hand side of the equation and call it a pseudoforce. If you've ever constructed Newton's laws in a non-inertial frame, this is what you probably did.
In general curved spacetime, you can't escape the $\Gamma$s so your options are limited to 2 and 3. When the Wiki article says that you don't need pseudoforces, it means that you can choose option 2, not that you can ignore them outright.
Finally, though in GR you cannot make the $\Gamma$s vanish everywhere, you can always make an instantaneous choice of coordinates to make them vanish at a point. As a result, at any given instant of time you can choose coordinates such that $f^\mu_{\mathrm{pseudo}} \equiv -m \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0$. Because $f^\mu_{\mathrm{pseudo}}$ can be set to zero by a coordinate transformation, we know that it is not a 4-vector because if a 4-vector vanishes in one coordinate system, it must vanish in all coordinate systems. This is the mathematical distinction between real forces, which are 4-vectors and therefore coordinate-independent geometrical objects, and pseudoforces, which can be viewed as artifacts of your choice of coordinates.
