Given a single cover $\mathcal{W}=\{W_i\}_{i\in I}$ of a space $X$ by nonempty sets, the nerve complex $\mathcal{N}(\mathcal{W})$ has $I$ as its vertex set, and it has $[i_0,\ldots,i_k]$ as a $k$-simplex if $\cap_{j=0}^k W_{i_j}\neq\emptyset$.

To see if I have understand your definitions correctly, is it correct to say that your complex in (2) is the nerve complex $\mathcal{N}(\mathcal{U}\cup\mathcal{V})$, where $\mathcal{U}\cup\mathcal{V}$ is the cover of $X$ given by $\mathcal{U}\cup\mathcal{V}=\{U_a,U_b,\ldots\}\cup\{V_1,V_2,\ldots\}$?

Also, is it correct to say that your complex in (1) is the nerve complex $\mathcal{N}(\mathcal{U}\cap\mathcal{V})$, where $\mathcal{U}\cap\mathcal{V}$ is the cover of $X$ given by $\mathcal{U}\cap\mathcal{V}=\{U_\alpha\cap V_j~|~U_\alpha\in\mathcal{U}\text{ and }V_j\in\mathcal{V}\}$? [For this nerve complex, if $U_\alpha\cap V_j=\emptyset$, then it would not be included as a vertex.]

Given a single cover $\mathcal{W}=\{W_i\}_{i\in I}$ of a space $X$ by nonempty sets, the nerve complex $\mathcal{N}(\mathcal{W})$ has $I$ as its vertex set, and it has $[i_0,\ldots,i_k]$ as a $k$-simplex if $\cap_{j=0}^k W_{i_j}\neq\emptyset$.

To see if I have understand your definitions correctly, is it correct to say that your complex in (1) is the nerve complex $\mathcal{N}(\mathcal{U}\cap\mathcal{V})$, where $\mathcal{U}\cap\mathcal{V}$ is the cover of $X$ given by $\mathcal{U}\cap\mathcal{V}=\{U_\alpha\cap V_j~|~U_\alpha\in\mathcal{U}\text{ and }V_j\in\mathcal{V}\}$? [For this nerve complex, if $U_\alpha\cap V_j=\emptyset$, then it would not be included as a vertex.]

Also, is it correct to say that your complex in (2) is the nerve complex $\mathcal{N}(\mathcal{U}\cup\mathcal{V})$, where $\mathcal{U}\cup\mathcal{V}$ is the cover of $X$ given by $\mathcal{U}\cup\mathcal{V}=\{U_a,U_b,\ldots\}\cup\{V_1,V_2,\ldots\}$?