Equivalence of Baire Space definitions

I am hoping someone could help me show that the following statements, which define a Baire Space, are equivalent.

Defn1: Any topological space X such that the intersection of any countable collection of open dense sets is dense is called a Baire space.

I know that this statement is equivalent to the following:

1)$(A_i)$ countable collection of closed nowhere dense sets $\rightarrow \operatorname{int}(\cup_i A_i) = \emptyset$.

2)Union of countable collection of closed sets has an interior point $\rightarrow$ one of the closed sets has an interior point.

3)Union of any countable collection of closed sets with empty interior has empty interior.

Now the issue is that I came across another definition of a Baire Space. It is as follows:

Defn2: $X$ is Baire iff every nonempty open set is nonmeager.

Are definitions 1 and 2 equivalent? If so, why?

Yes they are equivalent, because:

If $X$ is Baire then every non-empty open set is non-meager because a countable union of closed nowhere dense sets has an empty interior.

If every non-empty open set is non-meager, then no non-empty open set is contained in a countable union of closed nowhere dense sets, and thus a countable union of closed nowhere dense sets has empty interior and $X$ is Baire.