I am not entirely sure if there is a difference between $K$-rational points on a scheme $X$ over $k$ and $K$-valued points on $X$. Both seem to refer to a $k$-morphism Spec $K \to X$ but the difference is that in $K$-rational, $K/k$ is a field extension whereas in $K$-valued, $K/k$ might just be a containment of rings? Thanks.