Different definitions of complexity class DP
Equivalence follows immediately from the definitions.  The definition of $\setminus$ is:
$$A \setminus B = A \cap \overline{B}$$
where $\overline{B} = \Sigma^* \setminus B$ denotes the complement of $B$.  Now taking $L_1=A$ and $L_2=B$, it is trivial to prove that $L$ meets the first definition iff it meets the second definition, since $B \in NP$ iff $\overline{B} \in coNP$.