Let $(x_\alpha)_{\alpha\in I}$ be a set in $X$, where it is used $(I,\preceq)$ as a directed set. Which one of these definitions are correct, when we learn about the convergence of a net $(x_\alpha)_{\alpha\in I}$ in a topological space $X$?
Let $y\in X$. The net $(x_\alpha)_{\alpha\in I}$ is said to converge to $y$ if for every open set $U$ containing $y$ there exists $\alpha\in I$ such that ...
(1) $\quad$ ... $x_\beta\in U$ for all $\beta\in I$ with $\beta\geq \alpha$.
or
(2) $\quad$ ... $x_\beta\in U$ for all $\beta\in I$ with $\alpha \preceq \beta$.
I think it is reasonable to take (2). In literature, they use the first one, where they used $\leq $ instead of $\preceq$, so probably they meant $\geq$ as an relation, the reversed symbol of $\leq$, not the sense of usual order, I don't know. That's why I ask.
Just as with sequences we take the tail of a net, so
$$\forall U \in \mathcal{U}(y): \exists \beta \in I: \forall \alpha \in I: (\alpha \succeq \beta \to x_\alpha \in U)$$
so we can find an index such that all terms with a "larger" (in the sense of how $(I, \preceq)$ is defined) index are in $U$. For classic sequences we use $I=\Bbb N$ and $\preceq = \le$ as we know it for natural numbers, but the power of nets is that we allow any directed set, also ones that are much larger than just countable and linearly ordered. We can have partially ordered ones, etc. as long as they are directed: $\forall \alpha, \beta \in I \exists \gamma \in I: \gamma \succeq \alpha \land \gamma \succeq \beta$, so that tails always "come together again", providing some sense of direction, going "forwards"...
Your $(2)$ is the same as $(1)$ but there the tail starts at $\beta$ and $\alpha$ are the indices in the tail. It's more common to use $\succeq$ for the reverse of $\preceq$, just like $\le$ and $\ge$ are both used on $\Bbb N$.