I am trying to visualize the difference between immersed submanifold and embedded submanifold. At first, I thought that, for example, if I can embed manifold $M$ in $\mathbb{R}^4$ and if my friend can find an immersion $f:M\to \mathbb{R}^3$ then immersed $M$ in $\mathbb{R}^3$ is a shadow of embedded $M$ in $\mathbb{R}^4$. For checking my thought, I consider the Möbius strip. One can embed Möbius strip in $\mathbb{R}^3$, and immersed Möbius strip in $\mathbb{R}^2$ is just the limbered Möbius strip. It seems that what 2-dimensional creature sees about Möbius strip is shadow of Möbius strip on $xy$-plane ($\mathbb{R}^2$).(left figure)
So, if my thought is to be right and immersed $M$ in $\mathbb{R}^3$ is a shadow of embedded $M$ in $\mathbb{R}^4$, then I consider the real projective plane and say: the immersed real projective plane has different figures in $\mathbb{R}^3$ (Boy's figure, cross-cap figure and ...) because it has different shadows in different directions. Is that right? If no, why does real projective plane have different figures in $\mathbb{R}^3$? And which part of my idea is wrong?
Thanks in advance.