I'd be inclined to say that none is correct, although 3 is closest. The flux through an arbitrary surface ##S## is, as
@vanhees71 says, ##\int_S\vec E\cdot d\vec S##. If you consider a uniform field passing through a flat surface that simplifies to ##\vec E\cdot\vec A##, where ##\vec A## is the vector area of the surface, a vector perpendicular to the surface with magnitude equal to its area. So the flux is the component of the electric field perpendicular to the surface times the area (and the full integral formulation is just the same thing evaluated for each tiny bit of surface and summed over the surface).
Notice that I haven't mentioned field lines. Field lines are a way of visualising a field. Formally, they are
integral curves of the field, made by picking a point, taking a small step in the direction of the field vector there, and repeating (
this Insight on magnetic field lines has more detail). There are always infinitely many of them passing through any surface, which is why I said none of the three definitions in the OP is correct. However, we usually choose to draw a fairly regular set of field lines emnating from a simple surface, and how many of them pass through another surface is a reasonable approximation to the flux as long as they are pretty much perpendicular to the surface. When the lines aren't perpendicular to the surface then the approximation isn't really useful.
So I would take the third definition and modify it a bit:
3. Electric flux is the number of electric lines passing through any area as long as they all pass through all surfaces of interest perpendicularly. You could apply similar caveats to definition 1, though, which is what I would guess that
@Doc Al has in mind.