If all "solar systems" are: 
'a star and all of the planets, comets, etc. which revolves around it.'

And it cannot be "Solar System" or "The Sun Solar System" because we are using these words to describe billions of other systems.
 

Yet I am not that well read and think I MUST be wrong.

Kind Regards to all!

Think of it this way: each object has a classification, a name, and perhaps a nickname or generic name.

So, the Sun is a particular star (our star). The word "sun" can also be applied generically, just like "star" to mean any star ... just not our star, whose name is the Sun (a proper noun). Just as there are many stars in the Milky Way, you could say there are many suns. But if you were to say the Milky Way contains many Suns, you would be implying "many stars like our Sun: perhaps as to spectral class, harboring a system of planets, etc."

Similarly, any star can be part of a solar system, but ours is called the Solar System (its name, a proper noun).

Gliese 581 (again, a proper noun) is a particular star, or sun, but it's not Sol, the Sun, our star. It apparently has its own solar system, but it's not the Solar System, just one among many solar systems.

Solar is not a proper noun. It means related to the Sun, but could be applied to any other star within the context of its own neighborhood. You could say that Gliese 581 is known to harbor a planet in its solar neighborhood, or solar system, but not its Solar System (that one belongs to us, or more properly to the Sun). It would be less confusing to refer to the neighborhood of Gliese 581as a planetary system (though that might tend to leave aside asteroids, comets, and the like).

Welcome to the discussion group, SomeDude.

Solar is the adjective we use in English to refer to the Sun even though it is of Latin origin.  Similarly, lunar is derived from the Latin adjective for referring to the Moon.  The term solar system originally referred to a mathematical system for predicting the positions of planets based on the assumption that the Sun is at the center.  It was synonymous with the term heliocentric system.  Eventually people started using the term solar system to refer to a specific set of objects, even though a system is actually a way of doing something.

In the past the names Sun and Moon were usually capitalized.  For much of the twentieth century grammarians recommended they not be capitalized.  I never understood that.  More recently there has been a trend toward capitalization.  That is what I do.  I generally do not capitalize solar system, but I have no quarrel with anyone who does.

I am definitely not a pro, but, from what I do know....Sun is the name of the star our planet orbits.  I am going to guess that other cultures have other names for it, just as inhabitants (if the exist) of planets outside our solar system would have a name for their star.  It's rather hard to say what any other sun would be named, because so many factors influence the names humans give to specifc things.  The solar system may or may not have it's own name.  As far as our solar system is concerned, I do not know if it has a proper name or not.

I have always interpreted "Sun" as referencing the star that provides Earth's heat and light, and "sun" to mean stars that are like our sun.  Likewise "Moon" would mean Earth's natural satellite, while "moon" would be for the natural satellites of other planets, ie. Titan is one of Saturn's moons.  I find it ...interesting...that anyone would recommend not capitalizing the first letter of Sun and Moon when used as a proper noun.  The things one learns.

Maybe we will use the system that Star Trek did for naming.

 Kilanath is correct here. No offence Centaur. But Sun and Solar System refers to our own little arrangement in the local cosmos area. solar system would be applied to other 'sun systems.'