ElGamal with elliptic curves II
I have a concern regarding the security of the scheme -- let's suppose that there are only two possible messages, $m_0$ and $m_1$. Then, the encryption is done by multiplying, in the field, the chosen message with the $x$-coordinate of an unknown point on the curve.


What I highlighted is your problem; that is an invalid way of combining the point which is the result of the DH exchange with the plaintext.

In the standard El Gamal scheme, we use the 'modular multiplication' operation as the group operation within the key exchange, and hence we use that to combine the plaintext with the DH value.

So, when perform the operations on an Elliptic Curve, we do the same.  So, what you need to do is (somehow) convert the plaintext into an elliptic curve point, and add it to the point derived from the DH exchange; if the plaintext point is in the same prime-sized subgroup is the DH generator, you're safe.

Alternatively, you can just do ECIES, and not have to worry about translating plaintexts into points...