You've made a misunderstanding there. For your rotating coil, the charges in the coil are already moving. They're rotating. What actually happens is that the changing magnetic flux produces an electric field, which only THEN acts on the charges. The magnetic field itself doesn't actually interact with the charges.
By the Lorentz force formula, F = qv cross B, if v=0, F=0. Magnetic fields only interact with moving charges.
However, the distinction between an electric and magnetic field is a fine one because special relativity would then go on to say they're the same thing. They ARE similar to some extent. Imagine you have an electron moving in a straight line at constant speed. Let's say you have two observers, observer A who is in the lab frame and observer B who is moving with the electron. Now put this electron in a 'magnetic field'. The observers aren't accelerating with respect to one another, so they must measure the same forces. Observer A sees a moving electron acted upon by a magnetic field, as expected. Observer B must therefore also measure some form of disturbance to the electron's path. However, observer B is moving WITH the electron; he/she sees the electron as stationary. Therefore, observer B can't be seeing a magnetic force acting on the electron. As a result, observer B sees an electric force acting on the electron.
Confusing? This is why VCE physics only scrapes the surface of things.