All of my geometric intuition for "immersed" versus "embedded" surface is contained in my geometric intuition for "immersions" versus "embeddings". In particular, as many people have pointed out, immersions need not be injective. But, of course, even injective immersions need not be embeddings. As a very simple example, consider the map $f : (-\pi, \pi) \to \mathbb{R}^2$ given by
$$
f(t) = (\sin t, \sin 2t).
$$
The image of this map is a sort of "figure 8" in the plane, traced out starting at the origin, moving through quadrants II, III, I, and IV, in that order, as $t$ moves from $-\pi$ to $\pi$. It's easy to see that $f$ is an injective immersion, but $f$ is not an embedding, since every neighborhood of $\mathbb{R}^2$ containing the origin also contains points of the form $f(-\pi + \epsilon)$ and $f(\pi - \epsilon)$ for all sufficiently small $\epsilon$.
As another one-dimensional example of this type, you could consider the closed topologist's sine curve with a loop, which is the graph of $g(x) = \sin(1/x)$ for $x \in (0, 1]$ together with the $y$-axis between $y = -1$ and $y = 1$ together with a "loop" smoothly connecting the point $(0, -1)$ to the point $(1, \sin(1))$. It's clear that there is some injective immersion $f : [0, \infty) \to \mathbb{R}^2$ whose image is this curve, and this immersion is not an embedding.
You can, of course, easily make either of these example into a surface by considering $h: (-\pi, \pi) \times (0,1) \to \mathbb{R}^3$ given by $h(t, s) = (f(t), s)$.
On the other hand, I don't really know any examples of injective immersions of surfaces which aren't embeddings that are "interesting" in a way that's fundamentally different from the examples above. The idea I have is that immersions are allowed to "approach themselves" or "limit onto themselves" in crazy ways that embeddings are not. In particular, if $f : X \to Y$ is an injective immersion, the topology on $f(X)$ as a subspace of $Y$ might be very different than the topology on $X$.
This is only a topology problem. Since that $M\subset \mathbb{R}^{3}$, $M$ have a relative topology induced by the topology on $\mathbb{R}^{3}$, $i.e.$, the open sets (or open neighborhood) $V$ on $M$ are open subsets on $\mathbb{R}^{3}$ intersection $M$ ($V=U\cap \mathbb{R}^{3}$). If $\chi: D\rightarrow M$ is a proper patch then this is a continuous function one-to-one with continuos inverse (topologically speaking), then $\chi$ is a homeomorphism between $D$ and $M$. Then $\chi(E)$ is open if $E\subset D$ is open. Another way is, W.L.G. $\chi(E)\subset \chi(D)\cap\rho(F)$ with $\rho:F\rightarrow M$ other proper patch. By your corolary $(\rho^{-1}\circ\chi)(E)$ is open since $\rho^{-1}\circ\chi$ is a difeomorphism on $\mathbb{R}^{3}$ and then $\rho\circ(\rho^{-1}\circ\chi)(E)$ is open on $M$, but $\rho\circ(\rho^{-1}\circ\chi)(E)=\chi(E)$ since $\rho$ and $\chi$ are one-to-one.
Best Answer
Lipschutz's definition of a simple surface is equivalent to the definition of a regular surface in doCarmo's book Differential Geometry of Curves and Surfaces. The latter might be a better starting point to learn differential geometry with a more standard & modern terminology.
In chapter 5.10 of DoCarmo's book he compares several further notions of surfaces, starting with that of an abstract surface, which is the same thing as a smooth $2$-dimensional manifold. In summary, as elaborated on on page 441:
The notion of immersion/embeddings is modern terminology and essentially boils down to allowing self-intersections (for immersions) and not allowing them (for embeddings). So e.g. the typical picture of a Klein bottle you may be familar with is an immersed surface (it has self-intersections) but not an embedded surface. It does not meet Lipschutz's second criterion (2), because at a self-intersection, a particular coordinate patch will only cover one branch, while the intersection of open subset with the surface must contain parts of both branches.
As a sidenote: There is also a more modern usage of the term simple surface, which is a lot more restrictive. (It is a Riemannian surface with strictly convex boundary, such that any two points are connected by a unique geodesic, depending smoothly on the endpoints).