According to the definition of electric flux, only the normal components of field lines passing through a surface are considered. I am unable to understand why it's so? What would be different or wrong if the definition included field lines in any direction?
Why does electric flux only consider normal components of electric fields
electric-fieldsgauss-law
Related Solutions
You're not particularly spot on with your definition of electric flux. Most fundamentally, the electric flux $\Phi$ through a given surface $S$ is defined to be $$ \Phi=\int_S \mathbf E\cdot\mathrm d \mathbf S. $$ If you introduce a well-defined model in terms of field lines, then this does end up describing the number of field lines that cross $S$, to within the limits of the model. When the model is accurate, both share the same features: saying that $\Phi$ depends on the number of field lines instead of their density is the same as saying that $\Phi$ stays constant if we increase the decrease the line density and proportionally increase the surface area, and indeed the same is true if you decrease the electric field strength and proportionally increase the surface area.
The reason we define the electric flux is because it is useful. It is precisely the correct quantity to relate the electric field to the existing charges, and this is done via Gauss's law, $$ \oint_S \mathbf E\cdot\mathrm d \mathbf S=\frac{1}{\epsilon_0}Q_\text{enc}. $$ This is the fundamental law of electrostatics, really, and it all flows from here (and also the superposition principle). What else do you need for it to be physically meaningful?
Another misconception is to say that the electric flux
is not proportional to the relative density of field lines, which would supply information regarding the strength of the field at that point.
It cannot tell you anything about what happens at any given point because it's not a function of any point, it talks about what happens on a given surface as a whole. And, if you're given a space-dependent vector field and a surface, there's not really many invariant ways to combine them other than through the flux.
The formal definition of the flux of a vector field $\mathbf E$ through some surface $S$ is given by
$$\iint_S\mathbf E\cdot\text d\mathbf a$$
where $\text d\mathbf a$ is a vector of magnitude equal to the area $\text da$ and direction normal to the surface $S$.
So yes, you do need to consider the direction of the field.
The "number of field lines" description is a conceptual way to understand it, but that is about as far as it goes. As you note, this doesn't take direction into account. Also, you have to define what your "number of field lines" means, as the number of field lines one decides to uses is subjective.
Best Answer
Only vector components that are normal to a surface will actually pass through it.
For an oblique vector and a surface, only the normal component of the vector "passes through". The in-surface component is inconsequential.