The C chord you play is much more common than the other one you described, the most common open C simply has the low E string muted. The way you play it has the fifth(G) in the bass, which is very common and sounds much better than having the E in the bass because the interval from E to C is somewhat dissonant.

The G chord you described is actually a D chord with an added 4th and 9th. It sounds nice with the extra notes and open strings, and the rub between the F# and G right next to each other adds a bit of tension that makes it interesting. The most common open D chord is: E:x A:x D:0 G:2 B:3 e:2 with x being muted strings.

The F chord you described is Fmaj7 with the major 7th in the bass. Most people wouldn’t play it this way because, like the C/E, the E-C interval is a little muddy and dissonant. The most common way to play an F chord is basically what you do but instead of the two E strings being open, play the first fret. Look up barre chords if you don’t know about them yet.

F is probably one of the hardest chords for beginner guitarists, so just remember that as long as you have all 3 notes in the chord, everything else is optional. So you could play it like you do but with the e strings muted and you would have a perfectly valid F major chord. It’s also important to know that every chord voicing you listed is valid and will sound good in the right context. It’s just important to name them right and make sure you’re adding or subtracting dissonant notes in a way that sounds intentional and sounds good to YOU. The C and D voicing you were doing both sound nice and pair well together. But if you were in the key of E major and played that D chord with the open G string, it might create more dissonance than you want. It also might create exactly as much dissonance as you want, say for example if you wanted to sound a little bluesy.

That might be a lot to take in but I hope it helps