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For these type of files, use your same first equation, but in this form:

$$V(r,\phi,\theta)=-{\frac{GM}{r}} +\sum _{n=2}^{\infty}\sum _{m=0}^{n}{\frac {P_{n}^{m}(\sin \phi )(C_{n}^{m}\cos m\theta +S_{n}^{m}\sin m\theta )}{r^{n+1}}}$$

Note that the second summatory now begins at $m=0$. Zonal harmonics are $C^0_{n}=J_n$ and $S^0_{n}=0$, $n=2...\infty$.

For these type of files, use your same first equation, but in this form:

$$V(r,\phi,\theta)=-{\frac{GM}{r}} +\sum _{n=2}^{\infty}\sum _{m=0}^{n}{\frac {P_{n}^{m}(\sin \phi )(C_{n}^{m}\cos m\theta +S_{n}^{m}\sin m\theta )}{r^{n+1}}}$$

Note that the second summatory now begins at $m=0$. Zonal harmonics are $C^0_{n}=J_n$ and $S^0_{n}=0$, $n=2...\infty$.

Clarification: Note that the answer equation is the same the OP has written but putting the zonal harmonics term within the series expansion. If the OP wants to use its equation he just has to take $J_n=C^0_n$ from the file he is referring

For these type of files, use your same first equation, but in this form:

$$V(r,\phi,\theta)=-{\frac{GM}{r}} +\sum _{n=2}^{\infty}\sum _{m=0}^{n}{\frac {P_{n}^{m}(\sin \phi )(C_{n}^{m}\cos m\theta +S_{n}^{m}\sin m\theta )}{r^{n+1}}}$$

Note that the second summatory now begins at $m=0$. Zonal harmonics are $C_{n0}=J_n$ and $S_{n0}=0$, $n=2...\infty$.

For these type of files, use your same first equation, but in this form:

$$V(r,\phi,\theta)=-{\frac{GM}{r}} +\sum _{n=2}^{\infty}\sum _{m=0}^{n}{\frac {P_{n}^{m}(\sin \phi )(C_{n}^{m}\cos m\theta +S_{n}^{m}\sin m\theta )}{r^{n+1}}}$$

Note that the second summatory now begins at $m=0$. Zonal harmonics are $C^0_{n}=J_n$ and $S^0_{n}=0$, $n=2...\infty$.

For these type of files, use this $$V(r,\phi,\theta)=-{\frac{GM}{r}} +\sum _{n=2}^{\infty}\sum _{m=0}^{n}{\frac {P_{n}^{m}(\sin \phi )(C_{n}^{m}\cos m\theta +S_{n}^{m}\sin m\theta )}{r^{n+1}}}$$ which is the same as the formula you had, but note that the second summatory begins at $m=0$. Zonal harmonics are $C_{n0}=J_n$ and $S_{n0}=0$, $n=2...\infty$.

For these type of files, use your same first equation, but in this form:

$$V(r,\phi,\theta)=-{\frac{GM}{r}} +\sum _{n=2}^{\infty}\sum _{m=0}^{n}{\frac {P_{n}^{m}(\sin \phi )(C_{n}^{m}\cos m\theta +S_{n}^{m}\sin m\theta )}{r^{n+1}}}$$

Note that the second summatory now begins at $m=0$. Zonal harmonics are $C_{n0}=J_n$ and $S_{n0}=0$, $n=2...\infty$.