Constructing OR gate with OT

I am constructing a two-party OR gate and trying to do this with oblivious transfer. Yet I am very new to oblivious transfer, wishing to know whether the following construction makes sense.

Goal: Alice inputs a random bit $a$, Bob inputs a random bit $b$, and Bob outputs the logical OR bit $a\oplus b$.

Construction via 1-out-of-2 OT: Alice inputs $m_0 = a, m_1 = 1$, Bob inputs $c = b$, and finally Bob returns $m_c$. Obviously, Bob returns $0$ iff $a=b=0$ and returns $1$ otherwise.

My first confusion is could I fix $m_1$ as a constant? Secondly, the probability of Bob knowing $a$ is $1$ when $b=0$ and $\frac{1}{2}$ when $b=1$, respectively. It seems the output (i.e., $a \oplus b$) leaks Alice's secret $a$ in a combinatorial probability of $\frac{3}{4}$. Should it be a valid OT?