That's understandable. We grow up playing chords in root position, which means that the name of the chord is on the bottom, with the other two notes an interval of a 3rd above each other. (E is a 3rd above C, and G is a 3rd above E). For example, when we play the C chord in root position, C is the lowest note in the chord, so it seems obvious that it is the C chord.
But when we see the C chord with E on the bottom, or G on the bottom, it's not so obvious, partly because the chord is no longer a stack of 3rds.
Chords upside down are called "inversions".
Here's the deal:
Every 3 note chord (called a "triad" -- trio -- tricycle -- meaning "3") can be played in 3 different positions -- inversions:
*Root position = The name of the chord is the bottom note
*1st inversion = The name of the chord is the top note
*2nd inversion = The name of the chord is the middle note
So when C is the lowest note of the C chord, it is called "root position". When C is the top note of the C chord, it is called "1st inversion". And when C is the middle note of the C chord, it is called "2nd inversion".
So a root position triad (a triad is a 3-note chord) is a stack of 3rds; actually, a minor 3rd on top of a major 3rd. A first inversion triad is a stack with an interval of a 3rd on the bottom and a 4th on top. A second inversion triad is a stack with an interval of a 4th on the bottom and a 3rd on top.
Here's what: Each inversion has it's own sound, so you can get a variety of sounds by using one inversion and then another. Each inversion also has its own feel, so some pianists find it easier to use a particular inversion than others, particularly to move smoothly from chord to chord.
So what happens when there are more than 3 notes in a chord, as in a 6th chord or a 7th chord?
Same deal -- it's just that now there are 4 positions of the chord instead of 3 as in a triad; root position, 1st inversion, 2nd inversion, and 3rd inversion. That gives the pianist lots of choices for voicing and fingering.
There's no law, either, that a pianist has to use all the notes of a given chord. If I want a more open sound, I might leave out the 5th of a 4-note chord, and just use the root, 3rd, and whatever the other note is -- 6th, 7th, major 7th, 9th, or whatever.
For example, I might voice a C7 chord with E on the bottom, skip the G, then include the Bb and C. Or I might play it as an arpeggio (broken chord) by playing a low root an octave lower, then play the 5th, then the 3rd an octave higher, and then come back to the Bb.
The choices are almost infinite, and the more complex the chord, the more exciting voicing choices there are.
So don't settle for just one position of a chord -- stand it on it's head and experiment with all the luscious choices for voicing it to create a sound all your own.