Deriving the effective potential from the Schwarzschild metric

I am attempting to derive the effective potential for a test particle around a massive object from the Schwarzschild metric; however, an extra term has appeared in my solution compared to Wikipedia’s, which I am most confused about. A quick summary of my derivation is below; why is this extra term here, and what is its physical meaning?

My Method:
Assuming the particle is on the equatorial plane where $phi=frac{pi}{2}$, the Schwarzschild metric becomes $$-c^2mathrm{d}tau=-left(1-frac{r_s}{r}right)c^2mathrm{d}t^2+left(1-frac{r_s}{r}right)^{-1}mathrm{d}r^2+r^2mathrm{d}theta^2$$
The radial equation is $$ddot{r}=-Gamma^1_{munu}dot{x}^mudot{x}^nu$$
where an overdot represents differentiation with respect to the proper time. The appropriate Christoffel symbols are $$
begin{matrix}
Gamma^{1}_{00}=frac{c^2r_s}{2r^2}left(1-frac{r_s}{r}right) & Gamma^{1}_{11}=-frac{r_s}{2r^2}left(1-frac{r_s}{r}right)^{-1} & Gamma^1_{22}=-rleft(1-frac{r_s}{r}right)
end{matrix}
$$
making the radial equation $$ddot{r}=-frac{c^2r_s}{2r^2}left(1-frac{r_s}{r}right)dot{t}^2+frac{r_s}{2r^2}left(1-frac{r_s}{r}right)^{-1}dot{r}^2+rleft(1-frac{r_s}{r}right)dot{theta}^2$$
From the metric, we know that $$dot{t}^2=left(1-frac{r_s}{r}right)^{-1}left(1+frac{1}{c^2}left(1-frac{r_s}{r}right)^{-1}dot{r}^2+frac{r^2}{c^2}dot{theta}^2right)$$
Substituting this in, we get $$ddot{r}=-frac{c^2r_s}{2r^2}+rdot{theta}^2-frac{3r_s}{2}dot{theta}^2$$
Substituting in the definition of angular momentum $dot{theta}=frac{L}{mu r^2}$, where $mu$ is the reduced mass, multiplying across by the particle’s mass $m$, and substituting in the definition of the Schwarzschild radius $r_s=frac{2GM}{c^2}$, one finally gets $$mddot{r}=-frac{GMm}{r^2}+frac{mL^2}{mu^2 r^3}-frac{3L^2G(M+m)}{mu c^2r^4}$$ from which the potential is found to be

$$ bbox[5px,border:1.5px solid black]
{
U(r)=-frac{GMm}{r}+frac{L^2}{2mu r^2}+color{red}{frac{m}{M}frac{L^2}{2mu r^2}}-frac{G(M+m)L^2}{r^3mu c^2}
}
$$

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Why is the term in red there? It should be very small, so perhaps it is negligible? Is there a fundamental error in my derivation? If this is correct, what is the physical meaning of this term; is it a relativistic correction to the centrifrugal force?