Since no answers have been forthcoming I will summarise what has been discussed in the comments.

The usual analysis of geodesic motion around a spherical mass assumes that the spacetime geometry is described by the Schwarzschild metric, and that the test mass is too small to perturb this metric to any significant extent. In that case the motion can be analysed using an effective potential that includes the gravitational forces and the fictitious centrifugal force. As described in the link you cite this produces an effective potential:

$$ V_{eff}(r) = -\frac{GMm}{r} + \frac{L^2}{2mr^2} - \frac{GML^2}{c^2mr^3} \tag{1} $$

(Carroll actually gives you the potential per unit mass, i.e. divide through by $m$, and he uses the usual relativist's units of $c = 1$.)

In a two body problem where the test mass is large enough to significantly perturb the system it's usual to describe the motion using the reduced mass $\mu$ given by:

$$ \mu = \frac{Mm}{M+m} $$

In the limit of $M \gg m$ this simply reduces to $\mu \approx m$. If you look at the effective potential described in the Wikipedia article it gives the potential without the assumption that $m$ is negligably small and uses the reduced mass $\mu$:

$$ V_{wiki}(r) = -\frac{GMm}{r} + \frac{L^2}{2\mu r^2} - \frac{G(M+m)L^2}{c^2\mu r^3} \tag{2} $$

If we take the limit $M \gg m$, so that $\mu\approx m$ and $M+m\approx m$, the equation (1) from Carroll and (2) from Wikipedia become identical.

I have to confess I'm not sure how the equation Wikipedia cite is derived, other than possibly as a heuristic approach. I don't think there is an analytical approach that takes into account the perturbation of the Schwarschild metric by the test mass.

Since no answers have been forthcoming I will summarise what has been discussed in the comments.

The usual analysis of geodesic motion around a spherical mass assumes that the spacetime geometry is described by the Schwarzschild metric, and that the test mass is too small to perturb this metric to any significant extent. In that case the motion can be analysed using an effective potential that includes the gravitational forces and the fictitious centrifugal force. As described in the link you cite this produces an effective potential:

$$ V_{eff}(r) = -\frac{GMm}{r} + \frac{L^2}{2mr^2} - \frac{GML^2}{c^2mr^3} \tag{1} $$

Note that Carroll gives the energy for unit mass, i.e. $m=1$:

In (7.47) we have precisely the equation for a classical particle of unit mass and "energy" $ {1\over 2}$E2 moving in a one-dimensional potential given by V(r).

and he uses the usual relativist's units of $c = 1$.

In a two body problem where the test mass is large enough to significantly perturb the system it's usual to describe the motion using the reduced mass $\mu$ given by:

$$ \mu = \frac{Mm}{M+m} $$

In the limit of $M \gg m$ this simply reduces to $\mu \approx m$. If you look at the effective potential described in the Wikipedia article it gives the potential without the assumption that $m$ is negligably small and uses the reduced mass $\mu$:

$$ V_{wiki}(r) = -\frac{GMm}{r} + \frac{L^2}{2\mu r^2} - \frac{G(M+m)L^2}{c^2\mu r^3} \tag{2} $$

If we take the limit $M \gg m$, so that $\mu\approx m$ and $M+m\approx m$, the equation (1) from Carroll and (2) from Wikipedia become identical.

I have to confess I'm not sure how the equation Wikipedia cite is derived, other than possibly as a heuristic approach. I don't think there is an analytical approach that takes into account the perturbation of the Schwarschild metric by the test mass.

Since no answers have been forthcoming I will summarise what has been discussed in the comments.

The usual analysis of geodesic motion around a spherical mass assumes that the spacetime geometry is described by the Schwarzschild metric, and that the test mass is too small to perturb this metric to any significant extent. In that case the motion can be analysed using an effective potential that includes the gravitational forces and the fictitious centrifugal force. As described in the link you cite this produces an effective potential:

$$ V_{eff}(r) = -\frac{GMm}{r} + \frac{L^2}{2mr^2} - \frac{GML^2}{c^2mr^3} \tag{1} $$

In a two body problem where the test mass is large enough to significantly perturb the system it's usual to describe the motion using the reduced mass $\mu$ given by:

$$ \mu = \frac{Mm}{M+m} $$

In the limit of $M \gg m$ this simply reduces to $\mu \approx m$. If you look at the effective potential described in the Wikipedia article it gives the potential without the assumption that $m$ is negligably small and uses the reduced mass $\mu$:

$$ V_{wiki}(r) = -\frac{GMm}{r} + \frac{L^2}{2\mu r^2} - \frac{G(M+m)L^2}{c^2\mu r^3} \tag{2} $$

If we take the limit $M \gg m$, so that $\mu\approx m$ and $M+m\approx m$, the equation (1) from Carroll and (2) from Wikipedia become identical.

I have to confess I'm not sure how the equation Wikipedia cite is derived, other than possibly as a heuristic approach. I don't think there is an analytical approach that takes into account the perturbation of the Schwarschild metric by the test mass.

Since no answers have been forthcoming I will summarise what has been discussed in the comments.

The usual analysis of geodesic motion around a spherical mass assumes that the spacetime geometry is described by the Schwarzschild metric, and that the test mass is too small to perturb this metric to any significant extent. In that case the motion can be analysed using an effective potential that includes the gravitational forces and the fictitious centrifugal force. As described in the link you cite this produces an effective potential:

$$ V_{eff}(r) = -\frac{GMm}{r} + \frac{L^2}{2mr^2} - \frac{GML^2}{c^2mr^3} \tag{1} $$

(Carroll actually gives you the potential per unit mass, i.e. divide through by $m$, and he uses the usual relativist's units of $c = 1$.)

In a two body problem where the test mass is large enough to significantly perturb the system it's usual to describe the motion using the reduced mass $\mu$ given by:

$$ \mu = \frac{Mm}{M+m} $$

In the limit of $M \gg m$ this simply reduces to $\mu \approx m$. If you look at the effective potential described in the Wikipedia article it gives the potential without the assumption that $m$ is negligably small and uses the reduced mass $\mu$:

$$ V_{wiki}(r) = -\frac{GMm}{r} + \frac{L^2}{2\mu r^2} - \frac{G(M+m)L^2}{c^2\mu r^3} \tag{2} $$

If we take the limit $M \gg m$, so that $\mu\approx m$ and $M+m\approx m$, the equation (1) from Carroll and (2) from Wikipedia become identical.

I have to confess I'm not sure how the equation Wikipedia cite is derived, other than possibly as a heuristic approach. I don't think there is an analytical approach that takes into account the perturbation of the Schwarschild metric by the test mass.