Why are solar systems heliocentric So this may be an incorrect assumption, but from my knowledge solar systems are heliocentric and there are always suns. (I realize to be a solar system there must be a sun semantically.) But it seems to me that planets and stars are just bodies of mass, so are there cases of solar systems where the star is just one of the bodies of the system and orbits around a super massive planet? And similarly are there 'solar systems' without stars, where there are a group of planets orbiting a massive planet at the center of the system?
Generally, astronomical bodies rotate around their common center of mass, not the one or the other body.

If one of the bodies is much heavier, the center of mass is much nearer to its center, sometimes even inside the larger body. For example, Earth and Sun both rotate around a point that is much nearer to the Sun's center than to the Earth's center, as the sun is much heavier. Looking from the outside, it seems as Earth rotates around the Sun, while the Sun just wobbles a bit in sync. But the reason for that is exclusively the masses of the bodies, not their being-a-star or not-being-a-star.

Your theoretical example of a super-heavy planet and a lighter star would result in the star seemingly rotating around the planet; however, here you need to look at why a star is a star:
Stars are created by gravity that is compressing the mass to a level that starts fusion, and thereby a star. In your example, either the 'light star' wouldn't make it into a star (and be a light body/planet), or the 'super-heavy planet' becoming a much brighter star (or both of it). That reverses the roles effectively.

Looking at it this way, the bodies in a system rotate always around the center of the mass, which is near the largest mass, which will be the first star of the system (if any).
So your first constellation is not possible.

The second is well possible - if neither of the bodies has enough mass to become a star, they will be still rotating around each other the same way, with the heaviest body being nearest the center. Such a system would be dark and basically invisible from some distance. We wouldn't be able to find them.