Referring to the unit normal vector Is the unit normal vector or the normalized normal vector at a point on a surface the same as "the normal vector" at that same point (and surface)? Does saying, "the normal vector" imply it is normalized? Wolfram Mathworld says:
The normal vector at a point  on a surface  is given by

--> If this is the case, why don't people, when referring to a vector at a certain point on a certain surface, just say "the normal vector" instead of "unit normal vector" or "normalized normal vector"
 A: Talking about "a normal vector" is confusing in this context, because of the subtle distinction between normal, which means perpendicular, and normalized, which means of magnitude 1.
Better would have been to say: "The vector normal to the surface at point $(x_0, y_0)$ is given by ..." 
A: Normal means perpendicular. There are lots and lots of normal vectors. 
If your surface is the $xy$-plane then every vector parallel to the $z$-axis is normal to the plane and they are all possible choices for a normal vector. There are a one-parameter family of vectors which are normal to the plane at any given point; they are given by $\lambda {\bf k}$, where $\lambda \neq 0$.
The unit normal vector is a normal vector that has unit length. There are exactly two unit normal vectors. One pointing upwards and one pointing downwards. In the case of the plane they are $\pm{\bf k}$.
In general, if ${\bf n}$ is any normal vector, then a unit normal vector is given by
$${\bf \widehat n}=\frac{\bf n}{\| {\bf n}\|}$$
where $\|{\bf n}\|$ is the length of the vector ${\bf n}$. If your surface is parametrised by $z=\mathrm{f}(x,y)$ then
$${\bf \widehat n} = \frac{\mathrm{f}_x}{\sqrt{\mathrm{f}_x^2+\mathrm{f}_y^2+1}}{\bf i}+\frac{\mathrm{f}_y}{\sqrt{\mathrm{f}_x^2+\mathrm{f}_y^2+1}}{\bf j}+\frac{-1}{\sqrt{\mathrm{f}_x^2+\mathrm{f}_y^2+1}}{\bf k}$$
Why is the unit normal so important?
The way the normal moves around tells us about how the surface is curved. Think of a car aerial on a car driving over a hill. The aerial rocks back and forth as the car goes up and down the hill. The derivative (Weingarten map) tells us how the normal moves around. The problem is that if the normal is changing length then the derivative will pick that up too. Even if the surface is totally flat, if the normal is changing length then the derivative will say that the normal is changing and we might infer that the surface is curved. If you work with the unit normal then any changes the derivative picks up must be coming from the curvature of the surface.
