Doubt about Empty Set's definition as a Set.

Saying that all the elements of a set are distinct does not imply that there are any elements at all. It just says there are not two of the same element.

That definition of a set is the customary one in beginning courses. The objects are mathematical objects.

What "well defined" means in that definition is that you know what set you have precisely when you know what things it contains, independent of how it is described. So, for example, the set of solutions to the equation $3x = 6$ and the set $\{2\}$ are the same set.

Often we want to talk about a set even if we don't know how to specify its elements. For example, we might want to consider the set $S$ of all odd perfect numbers. That's a perfectly good definition of a mathematical set, even though no one knows whether there are any odd perfect numbers. In other words, $S$ might be an empty set.

In fact, $S$ might be the empty set, since there is just one empty set. That's because a set is known when you know what is in it, and any two empty sets contain exactly the same things, namely, none.

The "distinct" in the definition you quote does not mean "particular" it means (essentially) "different". So the sets $\{2\}$ and $\{2,2\}$ are the same set.

Very broadly speaking, given a set $X$ and a property $P$ depending on elements $x \in X$, you can define the set $$Y:=\{ x \in X \text{ such that }x \text{ satisfies the property } P \}$$

For instance, if you have already defined the set of all integers, you can define $\{ x \in \mathbb N : x>2\}$ (the set of all integers larger than $2$).

But if you chose a property that no element of the set $X$ satisfies, then you get the empty set.

For instance, you could define the empty set as $\{ x \in \mathbb N: x \text{ is both even and odd}\}$. This is a set, and it contains no elements.

If you want a more fundamental explanation of why this is an authorized defintition, you have to go back to ZF axioms. Modern mathematics is built on a finite set of axioms from which everything can be deduced; and the fact that the definition of sets I mentionned is well-defined is exactly one of the axioms, see https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory (axiom schema of specification).