The short answer is: yes, galaxies are really shaped like that. Even the one we're in is a spiral.
Your idea of a light from different parts of the galaxy may seem plausible on the surface but if you treid to think this through you'd soon see flaws in it.
Here a few things to think of:
1] If the light from nearer versus farther parts of a galaxy were causing us to view them differently, then it would make a BIG difference whether those galaxies were edge-on versus flat-on to us. But we see spiral galaxies from every conceivable angle, including edge-on.
For example: it could never result in the very famous Sombrero Galaxy, which we see edge-on:
or this spiral, seen edge-on:
2] It is true that light from the edge of a galaxy might be a few thousand years older than the light from the centre of the galaxy, but galaxies shine for billions of years. We see light from an entire galaxy
in one shot, even if some of that light is slightly older. A delay in some parts of the light would simply not
produce a spiral pattern.