I'm having trouble understanding how these 2 curves differ, i.e. how the points on these curves differ.
As I understand, for an algebraic number field $K$, a point on $X_1(N)(K)$ parameterizes a pair $(E,P)$, where $E$ is an elliptic curve with coefficients over $K$ and $P$ is a $K-$rational point on $E$ of order $N$. Furthermore, a point on $X_0(N)(K)$ parameterizes a pair $(E,C)$, where $E$ is an elliptic curve with coefficients over $K$ and $C$ is a cyclic subgroup of order $N$.
So if we have $(E,P)\in X_1(N)(K)$, then $(E,\langle P\rangle )$ is a point on $X_1(N)(K).$ Conversely, if we have $(E,C)\in X_0(N)(K)$ and $P$ is a generator of $C$, then $(E,P)\in X_1(N)(K)$.
How are these curves different? Am I missing something? Does maybe $P$ from $(E,P)\in X_1(N)(K)$ have to be $K-$rational and $C$ from $(E,C)\in X_0(N)(K)$ only has to be $Gal(\overline{K}/K)-$invariant?