Measure theory based definition of general position This is a soft question. I would like a cursory overview of how the precise notion of a "generic point" came to be in algebraic geometry. I don't know any algebraic geometry, so this maybe a hard ask. What I do know and may be relevant here is undergrad level analysis, topology, algebra, some galois theory, and  manifold theory. 
As I understand, there was a notion of "general position", which let us make statements like "two generic lines meet at a generic point". What I understand from this is that the subset of pairs of parallel lines is somehow "much smaller" than the full space of all pairs of lines.
So, if I had to formalize the idea of general position of two lines, I would do it like this:
Consider the set of equations $E = \{ \mathbb R[x, y] \times \mathbb R[x, y] \} $. 
We consider a function which maps elements of $E$ to the number of solutions they have, called $C$ (for count). $C: E \rightarrow \mathbb N\cup \{ \infty \}; C(p, q) = |\{ (x_0, y_0) | p(x_0, y_0) = q(x_0, y_0)\}|.$ 
Next, we create a function called $M: \mathbb N \rightarrow \mathbb R^+ $ (for measure), which measures how many elements of $E$ have number of solutions equal to the input. That is, $M(c) = \mu(C^{-1}(c))$. [Caveat: I don't know how to give a measure $\mu$ on the set $E$, but I presume there is some way to do so?]

An element $(p, q) \in E$ are said to be in special position iff $M(S(p, q))$ is zero. An element not in special position is said to be in general position 

This matches the intuition that a set of equations are in general position, if the number of solutions they have is the common case, and  a set of equations are in special position if the number of solutions they have is the uncommon case / measure zero. 
However, this is not the definition at all. Rather, we have a  topological definition:

A generic point of the topological space $X$ is a point $P$ whose closure is all of $X$, that is, a point that is dense in $X$.

Why is this chosen as the definition? I don't understand the above definition in terms of the Zariski topology. However, mine seems far more natural to me: we are stating that equations in general position have a solution set whose cardinality occurs very often --"in general"
Has the definition I propose above been studied? Is it flawed, because one can't give such a measure $\mu$? Are there relationships between this definition and the usual definition in terms of the Zariski topology?
 A: Very briefly, the original concept of a generic point originally came from the Italian school of Algebraic Geometry. It was more of a vague idea than a precise definition, and pretty much meant something true for "most" points. The Italian school did a lot of good work but later on suffered from issues with rigor - eventually, there were enough incorrect proofs and incorrect results around that the school kind of fell apart and it took Zariski and Weil most of the 1940s and 1950s to clean up the mess and put Algebraic Geometry back on firmer foundations. After this cleanup, the idea evolved and came up to something resembling a real definition - something being true "at a general point" meant that it was true for all points in some Zariski-dense open subset. 
The connection to generic points is that if we're working on an irreducible variety $X$ and we have condition which is reasonably nice, then if the condition is true at the generic point, it is true for a general point: there is a Zariski-dense open subset $U\subset X$ where our condition is satisfied. This usually goes the other way, too: if we have a nice enough condition satisfied on some Zariski-dense open subset, then it's also true at the generic point. So the slogan is that behavior at the generic point should be the same as behavior at a general point.
Formalizing this intuition of what's happening at most points using the generic point let us make a lot of arguments more rigorously and more elegantly. It's also a natural consequence of the framing of algebraic geometry in terms of the spectra of rings (generic points correspond to minimal primes), which has produced a pretty useful theory for the last ~70 years.

Now on to your specific example. Your definition for $E$ as the set of equations doesn't match what you'd want for two lines in the plane, so let's fix that. A line in the plane is given by an equation $ax+by+c=0$ where $a,b$ aren't both zero at the same time. We can observe that these equations are invariant under scaling $a,b,c$, so we get that the object parameterizing lines in the plane is $\Bbb P^2 \setminus \{[0:0:1]\}$. Fix a line $L_1$ with corresponding point $[r:s:t]\in \Bbb P^2\setminus \{[0:0:1]\}$. For another line $L_2$ with corresponding point $[a:b:c]$, the condition that these lines intersect at one point in the affine plane is exactly that $as-br\neq 0$, which is true on $\Bbb P^2\setminus \{[0:0:1]\}$ on a Zariski-dense open set (the complement of a smaller-dimensional subvariety). So for a fixed line, a general line intersects it in one point.

I'm now going to argue that the typical definition of "general points" recovers your definition in nice cases and is easier to deal with in uglier cases. In nice cases, like when we're working over a field $k$ which comes with a reasonable translation-invariant measure (think $\Bbb R,\Bbb C,\Bbb Q_p$ for instance), if we have a closed algebraic embedding of affine varieties $X\subset Y$ and $X$ is lower-dimensional than $Y$, then the measure of the $k$-points of $X$ inside the $k$-points of $Y$ is automatically zero. The proof of this is the same as the zero-set of an analytic function in $\Bbb R^n$ being measure zero. What this means is that things in special position in the usual sense are also in special position for your definition in these situations.
In other cases where we don't necessarily have a measure around, we have to build it. This is more difficult than just using the Zariski topology on our variety, which we already have access to. Plus, there are times when it's actually hard - if we want to generalize this to arbitrary schemes (just like we can do for the concept of a general point lying in a Zariski-dense open), it's a lot of work and not totally clear what we should be going for. As a result, it's not often done.
