Wikipedia gives the following Lagrangian for central force problem:
$$\mathcal{L}=\frac12 m \dot{\mathbf{r}}^2-V(r)$$
where $m$ is the mass of a smaller body orbiting around a stationary larger body. However, Taylor Classical Mechanics gives the different Lagrangian:
$$\mathcal{L}=\frac12 \mu \dot{\mathbf{r}}^2-V(r)$$
where $\mu$ is reduced mass between two bodies. I don't understand which one is correct.
Quite generally we can write $$\mathcal{L}=\frac12 m_1 \dot{\mathbf{r}_1}^2+\frac12 m_2 \dot{\mathbf{r}_2}^2-V(r)$$ where $\mathbf{r} = \mathbf{r_2} - \mathbf{r_1}$. Letting $\mathbf R$ denote the center of mass, i.e. $$(m_1 + m_2)\mathbf R = m_1 \mathbf r_1 + m_2 \mathbf r_2,$$ we can rewrite the Lagrangian as $$\mathcal{L}=\frac12(m_1+m_2)\dot{\mathbf R}^2+\frac12 \mu \dot{\mathbf{r}}^2-V(r).$$ The first term is a function of $\dot{\mathbf R}$ only, and its only contribution to the solution is a uniform translation of the center of mass. The remaining terms are the more interesting ones governing the relative motion of the masses. Wikipedia likely implicitly assumes $m_1 \gg m_2$, in which case $\mu \approx m_2$.
Edit: Upon reading the beginning of the relevant section of the Wikipedia article, the problem it describes is "a single particle with mass $m$ moving in a potential field $U(r)$", so it is effectively assuming one of the masses is kept fixed. The Lagrangian above describes the problem in which both masses are free to move.
