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@campfire said: “I tried playing an A-shaped barre chord with index barring 3rd fret, but that’s not a C chord,”……
You are correct. If your index only bars the third fret it is a Bflat chord
To make a C chord, index finger bars the third fret (all strings) and the ring finger bars D,G,B strings on the 5th fret. I am focusing on those 3 strings because they are the three strings that differentiate the A shape from the other shapes (mostly). Bar those three strings on the fourth fret to make B chord and since there is only a half step between B and C when you go a half step higher to the 5th fret that is where the C chord is made. Again, only focusing on DGB strings. Any of those bars can be made into a 6 string chord “A shaped chord” if you bar the rest of the strings two frets back.