The Néron model of $X$ is the smooth locus of the minimal regular model of $X$ (see Bosch-Lütkebohmert-Raynaud: Néron models, §1.5). The equivalence is then clear using the classification of Kodaira-Néron of the types of reduction of the minimal regular model.

Note that if $g(X)\ge 2$, then it is also true that $X$ has semi-stable reduction if and only if its Jacobian has semi-abelian reduction (Deligne-Mumford, based on Raynaud's description of Néron models of Jacobians).

If $g(X)=1$ but $X$ doesn't have a rational point, then the statement is no longer true. But one can show that the Jacobian of $X$ has semi-abelian reduction if and only if the type of the reduction of $X$ is a multiple of $I_n$, $n\ge 0$ (use the fact that $X$ covers its Jacobian. The additive reduction case is treated in a paper of Lorenzini, Raynaud and myself in 2004).

You don't need the completeness hypothesis on the base DVR.