This question is primarily regarding the definition of a generic point of a topological space that I came across in Qing Liu's Algebraic Geometry and Arithmetic Curves. First I will give the definition of a generic point that I am used to:
Definition I am used to: Let $X$ be a topological space. A point $x\in X$ is a generic point of $X$ if $x$ is dense in $X$. (In particular this implies that if a topological space has a generic point, then the space must be irreducible. So, reducible spaces do not have generic points.)
Definition in Qing Liu's book: Let $X$ be a topological space. Given $x,y \in X$, we say $x$ specializes to $y$ iff $y \in \overline{\{x\}}$. We say $x$ is a generic point of $X$ if $x$ is the unique point of $X$ that specializes to $x$.
From what I have been able to deduce so far, under Liu's definition of a generic point, reducible spaces can have generic points. For example if we take a ring $A$, then the generic points of $Spec(A)$ (according to Liu's definition) are all the minimal primes of $A$. And certainly, $Spec(A)$ is reducible unless $Nil(A)$ is a prime ideal.
Also, Liu's definition implies that if $O \subset X$ is a non-empty open subset and $x \in O$ is a generic point of $O$, then $x$ is a generic point of $X$. So, something that is locally a generic point is also globally a generic point. This is not true for the first definition.
So far I had been happily using the first definition of a generic point, but Qing Liu's definition has thrown me a little off-guard. I guess I do not fully understand the motivation behind Qing Liu's definition, and also the need for two different definitions that are clearly not equivalent. Which definition do people generally use?
Looking at Wikipedia it seems that the first definition of a generic point of a topological space is more prevalent. Any explanation or motivation for Qing Liu's definition will be useful, along with an explanation (perhaps?) of why there are two different definitions for the same concept.
Also, if answers are too long to type up, then references that explain the concept of generic point are more than welcome. But, I hope they address the questions I have asked.