In a scheme, each point is a generic point of its closure. In particular each closed point is a generic point of itself (the set containing it only), but that's perhaps of little interest. A point that's not closed, is probably more interesting, and there are no such thing in ordinary varieties.

What I have been wondering about is that there must be a good reason they are called generic points. Here are what I have got so far:

• A non-closed generic point is not closed, so it cannot be cut out from the scheme by any polynomial equations in any affine patch, and thus it does not posses any extra algebraic property that's not shared by others.
• A non-closed generic point is not a specialization of the scheme.

Are these correct? If not, what's the right intuition? Also, can the following statement be make more precise using the language of generic points?

• A degree $n$ and a generic degree $m$ algebraic curves intersect at $n \cdot m$ distinct points in $\mathbb{P}^2$ (the planar Bezout's theorem)
• Common solutions of $n$ polynomial systems in $n$ variables with generic complex coefficients in $\mathbb{C}^\ast$ are all isolated. (corollary of the Cheater's homotopy theorem)
• The number of common isolated solutions of $n$ polynomial systems in $n$ variables with generic complex coefficients in $\mathbb{C}^\ast$ equals the mixed volume of the Newton polytopes of the system. (Bernshtein's theorem)
• Generic points on a nonreduced scheme have isosingular structure (i.e., at all such points the local ring fail to be reduced in exactly the same way).

In each case I am familiar with their original meaning of the word generic, but I'm wondering if we can state the genericity conditions using the concept of generic points of schemes.