Alternative definition of product topology

That's equivalent to the usual definition.

The topology with a certain subset as a subbasis is the topology with the fewest open sets where all elements of that subset is open.

The product topology is the topology with the fewest open sets where every projection is continuous. That is: the preimage of any open set under the projection must be open. That is precisely the definition you give.

Equivalently, it's the topology generated by sets of the form $\prod U_i$, where $U_i=X_i$ for all but finitely many $i$. This can be contrasted with the "box topology", where any number of $U_i\ne X_i$.

The product topology makes the product $X=\prod X_i$ a "categorical product".

Also, the product topology satisfies a universal property, namely that any continuous map from another toplogical space to $X$ "factors through" the projections.