Asking "what is $\mathrm{Pr}(|{\Psi}\rangle=c_1|{1}\rangle+c_2|{2}\rangle$?" is not a well-defined question, because it does not correspond to a specific physical procedure.
However, you can ask something like
if I perform a projective measurement on the basis $|\phi_1\rangle = c_1|{1}\rangle+c_2|{2}\rangle$, $|\phi_2\rangle = c_2^*|{1}\rangle-c_1^*|{2}\rangle$, what is the probability that the measurement outcome will be $\phi_1$?
where the crucial differences are
- the question starts with "If I perform a projective measurement...", and
- the question asks for the probability of an outcome of that projective measurement.
Once you do that, if the system starts in the pure state $|\Psi\rangle$, then the probability that the measurement outcome will be $\phi_1$ is given by
$$
\mathrm{Pr}(\phi_1)
= \big|\langle \phi_1|\Psi\rangle\big|^2
= \left|\left(c_1^*\langle 1| + c_2^* \langle 2|\right)|\Psi\rangle\right|^2.
$$
Note, however, that this question is not equivalent to asking "Is the system in a superposition state?", as implied in your question title, particularly if you interpret the connector "a" to include all possible superposition states. (Where, moreover, the term "superposition states" needs to be specified as relative to a given basis. The term has no meaning of its own without that.) That question does not correspond to any specific projective measurement, so you cannot assign it to the Born-rule probability of any experiment. If you have a large ensemble of systems coming from the same preparation procedure, then you can perform a quantum-state-tomography procedure that will give you the exact density matrix of the system, including its purity, but even with that the question "what's the probability that the system is in a superposition?" still has no assignable meaning.
