A property between "separable" and "second countable"

Consider $\mathbb R$ with the finite complement topology. Any infinitely countable subset $A$ of $\mathbb R$ is dense since an open subset of $\mathbb R$ can only miss finitely many points of $A$.

Let $\mathcal B$ be a countable family of non-empty open subsets of $\mathbb R$. Then $\bigcap_{B \in \mathcal B} B$ misses countably many points of $\mathbb R$ at most. It follows that some $x \in \mathbb R$ lies in every element of $\mathcal B$. Thus, the open subset $\mathbb R - \{x\}$ does not contain any element of $\mathcal B$.

What you're looking for is the concept of a $\pi$-base (or pseudobase), i.e. a collection of non-empty (this matters!) open subsets $\mathcal{P}$ such that any non-empty open subset of $X$ contains a member of $\mathcal{P}$. (The collection is downward-dense in the poset $(\mathcal{T}\setminus\{\emptyset\}, \subseteq)$ is another way of putting it)

The minimal size of a $\pi$-base for $X$ is denoted $\pi w(X)$ (rounded up to $\aleph_0$ if necessary, in Juhasz it's $\pi(X)$) , see the cardinal functions sections of this wikipedia page. So property $C$ is countable $\pi$-weight or $\pi w(X)=\aleph_0$ in more conventional terms, and I believe property C is already taken as a name in topology, or at least property (K) is, for sure. (which has the related meaning that every uncountable set of open subsets has an uncountable subset that pairwise intersect; a property implied by but weaker than separability). I prefer countable $\pi$-weight, or having a countable $\pi$-base as a name, being a bit more descriptive.

As to examples: For an $X$ just $T_1$ but not higher, the cofinite topology on an uncountable $X$ is separable and does not have a countable $\pi$-base. A more advanced example (compact Hausdorff): $[0,1]^{\Bbb R}$ in the product topology is separable but has no countable $\pi$-base, as a counting argument involving basic subsets will reveal. That both examples are not first countable is no accident: if $X$ is both separable and first countable, the union of the local bases at the countable dense subset form a countable $\pi$-base, as is easily checked. For metric spaces, having a countable base, being separable and having a countable $\pi$-base are all equivalent.

There also exists the notion of a local $\pi$-base at $x$: a collection of non-empty open subsets of $X$ such that every neighbourhood of $x$ contains a set from it. This is related to notions like tightness at a point etc. We get a similar cardinal invariant of $\pi\chi(x,X)$ for the minimal size of such a collection, etc.