"So by this explanation, the F Major is the only Major Chord that works this way? I just thought it also worked to move the A Major towards the bridge to get a B Major, and moving that chord further up the neck to get a C, D or E, but it would appear that I was wrong in that regard..."

No, there is two major bar chord shapes. "main shapes"
1. Is the F major shape we already covered.
2. Lets start with the B chord. Your first finger in the 2nd fret covers A/D/G/B/E strings. Finger #3 4th fret covers D/G/B/E, that makes a B chord.

By moving that shape so your first finger is in the 3rd fret you make a C major chord, move the shape to the 5th fret, thats D major and so on up/down the neck.