It is said that if two elliptic curves $E_1, E_2$ defined over a finite field $K$ are isomorphic, then $E_1(K), E_2(K)$ groups are also isomorphic. But the converse is not true. Now,
By definition, when $E_1, E_2$ are isomorphic over $K$ then $E_1$ can be transformed to $E_2$ by admissible change of variables. If I am not wrong, this implies that a point $P$ on $E_1$ is transformed to a point $P'$ on $E_2$.
If $E_1(K), E_2(K)$ groups are isomorphic implies that there exist a bijective map between the two groups where the elements are points. So, the points are mapped again.
Since we are interested in the curve groups so, does it matter whether one take isomorphic curve or isomorphic group? Similarly, I don't understand what are the advantages of 1. over 2.?