If we alter the conventional definiton of infinity in projective geometry, which fundamental structures or theorems are affected?

In standard projective (P^2), the line at infinity is defined as ([X,Y,Z]) with (Z=0). However, other authors define it as ([X,Y,Z]) with (X=0). I feel some confusion. Can it be shown that this does not depend on these choices? Is the elliptic curve group structure preserved under projective transformations? Usually, the identity element of the elliptic curve group structure is the point at infinity.