If the empty set is a subset of every set, why write ... $\cup \{∅\}$?

It is because the emptyset $\emptyset$ is a subset of every set, but not an element of every set. It is $\emptyset\in S$ and you might want that to show, that the elements of $S$ define a topology.

Or to be more clear it is $\{1\}\neq\{1,\emptyset\}$. The set on the left has one element, the set on the right has two elements, with $\emptyset\in\{1,\emptyset\}$

The answer is: the given definition uses $\cup\,\{\emptyset\} $, not $\cup\,\emptyset $, so it adds the empty set as an element, not a subset of $S $.

Because the empty set $(\emptyset)$ is one thing, but what you have there is $\{\emptyset\}$, which is a different thing: it's a set with a single element (which happens to be the empty set).