By filters I will mean filters on some set $\mho$.

I define product of an infinite family of filters in two ways. I feel (by analogy with properties of Tychonoff product vs box product of topological spaces) that these two products are different, but don't know how to find a counter-example supporting their non-equality.

The first product of family $a_{i\in n}$ (where $n$ is an (infinite) index set) of filters is defined as the filter induced by the base $$\Pi_1 = \left\{ \prod_{i\in n} A_i \,|\, A\in\prod_{i\in n}a_i \right\}.$$

The second product is the filter induced by the base $$\Pi_2 = \left\{ \prod_{i\in n} B_i \,|\, m\text{ is a finite subset of $n$}, A\in\prod_{i\in n\setminus m}a_i, i\in m\Rightarrow B_i=A_i, i\in n\setminus m\Rightarrow B_i=\mho \right\}.$$

Prove that $\Pi_1 \ne \Pi_2$ for some $a$ (counter-example wanted).