How could I make a MAC two time secure?
Something like $t = a m + b + c m^2$ does not seem to work, as $m^2$ is not evenly distributed in $\mathbb{Z}_p$. (Am I missing something?)


You are missing something.  Given two solutions $(m_0, t_0), (m_1, t_1)$ to the equation $t = cm^2 + am + b$ (for unknown $a, b, c$), any third solution $(m_2, t_2)$ will hold with probability $1/p$ (assuming $m_2 \ne m_0, m_1$); if $p$ is large, this exactly satisfies the criteria you're looking for.  You can see that by fixing $m_2$ and considering the possible $t_2$ values between $0$ and $p-1$; for each such $t_2$ value, there is a unique set of $a, b, c$ values that make all three equations work, and as $a, b, c$ were generated randomly (so any no set has any higher probability than any other), this means that all possible $t_2$ values are equiprobable.

This is precisely the insight that makes Shamir Secret Sharing work.

It doesn't matter that $m^2$ isn't equally distributed.  If it really bothers you, you can perform the operations in $GF(2^n)$ rather than $GF(p)$. In $GF(2^n)$, $m^2$ is evenly distributed (but, as above, it's not actually a concern).