Is parametrized surface a surface? This is a conceptual question. First, I learned these two definitions during my class:
Definition 1: Parametrized surface is a map $q(u,v):U \to \text{R}^3$, where $U$ is open and $U \subset \text{R}^2$. Also, we require that $\frac{\partial q}{\partial u}$,$\frac{\partial q}{\partial v}$ be linearly independent. 
Then we went on and on, talked about a lot of other stuff, like Monge patch. Then we came across this definition:
Definition 2: A surface is a set $M \subset \text{R}^3$ such that $\forall p \in V$, $\exists \text{ open }  O\subset \text{R}^3  $ such that $p\in O$, and $M \cap O$ can be realized as a Monge patch.
This might be a silly questions, but I think defnition 1 claims that when we are talking about a "parametrized surface", we are talking about a function. Yet when we talk about general surface, we are talking about a subset in $\text{R}^3$?
 A: The simplest difference is that a parametrized surface includes the parametrization - i.e. it is a surface along with some standard coordinates $u,v$. A general surface can be given local coordinates anywhere, but not necessarily global ones, and none of those coordinate systems are part of the object itself. This is the same distinction as between curves and parametrized curves - a parametrized curve in the plane is a map $[a,b] \to \mathbb R^2$, while a curve in the plane is the image of any such map, which can be given many coordinate systems.
In the case of your definitions, there are some slight subtleties in addition to this distinction, which are topological in nature:
Your definition of a parametrized surface makes no reference to injectivity or embeddedness, so it allows self-intersecting or non-embedded surfaces. It also means that the topology of the surface is going to be (assuming no self-intersection) the topology of $U$, so using this definition things like the sphere are not parametrized surfaces - we have to leave at least one point out of the image. The advantage of this definition is computational ease - you can write any function on the surface in terms of the global coordinates $u,v$.
A surface (as in your second definition - I would call this an "embedded surface") is a subset  $\mathbb R^3$ that is locally a parametrized surface - Monge patches are a special case of parametrized surfaces. This definition is much more abstract - there are no global coordinates, and even locally there are many choices of coordinates, none of which are singled out as canonical. However, this definition allows richer topology (e.g. the sphere can be realized by using two patches), and rules out self-intersecting and non-embedded surfaces.
