A base generates an unique topology?

In your condition (2), it is important to point that we must have $x \in B_3$, such that $x \in B_3 \subset B_1 \cap B_2$.

Let's organize our thoughts.

You have fixed a topological space $(X, {\cal T})$. A collection ${\cal B}\subset {\cal T}$ is a basis if and only if satisfies conditions (1) and (2). End.

Now take a set $X$, without topology, and consider the power set $\wp(X)$. If you take ${\cal B}\subset \wp(X)$ and ${\cal B}$ verifies conditions (1) and (2), then there exists a unique topology on $X$ which has $\cal B$ as a basis. End.

Ok?

$\mathcal T$ and $\mathcal T'$ are not the topologies generated by base $\mathcal B$.