Can there be two different elliptic curve $E_{1}$ and $E_{2}$ and two different rational points $P_{1}$ and $P_{2}$ such that $P_{1}, P_{2} \in E_{1}$ and $P_{1}, P_{2} \in E_{2}$ but $P_{1} + P_{2}$ is a different point for $E_{1}$ and $E_{2}$. If so, is it easy for find an example?

Or given two different rational points $P_{1}$ and $P_{2}$ is there a unique elliptic curve $E$ such that $P_{1}, P_{2} \in E$

Thank you