Explaining the motivation behind two different definitions of a generic point 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.
 A: From a purely topological point of view (since I know virtually nothing of algebraic geometry), Qing Liu's definition is a natural generalization of the other one.
Let $X$ be a topological space. Write $x\preceq y$ iff $\operatorname{cl}\{x\}\subseteq\operatorname{cl}\{y\}$; clearly $\preceq$ is a preorder (or quasiorder) on $X$; if $X$ is $T_0$, which I believe is generally the case in your setting, $\preceq$ is a partial order. Qing Liu’s definition makes a point generic iff it is maximal in $\langle X,\preceq\rangle$: $x$ is generic iff $x\preceq y$ iff $x=y$. The other definition makes a point generic iff it is a maximum in $\langle X,\preceq\rangle$: for all $y\in X$, $y\preceq x$. Let me call these notions QL-generic and strongly generic for short.
Let $M=\{x\in X:x\text{ is }\preceq\text{-maximal}\}$; then $\big\{\operatorname{cl}\{x\}:x\in M\big\}$ is a partition of $X$ into closed, irreducible subsets. Thus, a QL-generic point $x\in X$ is strongly generic in $\operatorname{cl}\{x\}$, which is an irreducible component of $X$. My guess is that some people find that this extra generality makes some things easier to talk about.
A: I'll point out why you don't have to be worried about any of the things you listed. First thing, I see no reason why irreducible closed subsets can't have generic points according to Liu's definition. The definition is made for an arbitrary X, so consider $Y\subset X$ closed and irreducible as a new topological space with the subspace topology to get the generic point there.
I'm also not sure what the concern about reducible spaces is about. If you break $Spec(R)$ up into its irreducible components, then as you point out the minimal primes are the generic points of the these components (which are irreducible). There is still no generic point for the whole space $Spec(R)$ which is still consistent with the first definition, and moreover most people probably implicitly extend the definition of generic point in the first sense to work on irreducible components.
For the concern about "locally" being a generic point implies being a global generic point. Again, since you made the comment about irreducible already, this shouldn't be a problem for the first definition. If $X$ is irreducible, then any open set is dense, and hence if $x$ is a generic point of some open $O$, then even in the first definition it is a generic point of $X$. 
Edit: Removed the first sentence, because people were apparently not reading what I wrote and just commenting on the first sentence. The point I'm making is if you generalize the first definition in the obvious way to reducible spaces by working on irreducible components (note that I did say this in the post) or if you restrict the second definition to the case where you have a generic point in the first sense, then they become equivalent. So in that sense, there is no "fundamental difference" that seemed to be alluded to in the question.
