I am trying to learn topology from these lecture notes (or look here for a longer version).
I was having some trouble with the difference between the definition of a $T_0$-space and a $T_1$-space. Eventually, with the help of this question I figured out the difference between the two. However I wasn't able to figure out the difference between the definitions given in the lecture notes. In the notes, a $T_0$-space is defined as a topological space $(X,\tau)$ such that (and I quote):
Given $x,y\in X, x\neq y$, there is a $U\in\tau$ containing precisely one of the two points. (I.e. all points are distinguished by $\tau$.)
On the other hand, a $T_1$-space is defined as:
For any $x,y\in X$ with $x\neq y$ there is an open set $U$ such that $x\in U$, $y\notin U$.
Now, I would interpret these definition as follows: for a $T_0$-space I would write $$\textbf{Given } x,y\in X: x\neq y\exists U\in\tau: x\in U,y\notin U,$$ and for a $T_1$-space: $$\textbf{For any } x,y\in X: x\neq y\exists U\in\tau: x\in U,y\notin U.$$ At first glance, looking only at these epxressions, I would say these definitions are identical, and replace both 'Given' and 'For any' with a '$\forall$' symbol. I know the crux of the definition of the $T_1$-space is that you can switch $x$ and $y$, which means you can find, at least, two open sets; one containing only $x$ and one containing only $y$. This is not the case for a $T_0$-space, where we can only find (at least) one open set; containing only $x$ and not $y$. But, in the definition given above I see no reason why we can't switch $x$ and $y$, just as we did in the $T_1$ case. (Again, see this question.)
Am I misunderstanding the definitions? Is this (symbolic) interpretation of the definitions correct? And if so, what is the difference exactly between the words 'given' and 'for any' which make the definitions different?
Thanks so much in advance!