Is this explanation of electromagnetic induction correct? More specifically interpreting Lenz Law

When the magnet, starting with the north pole, goes towards a coil of wire (maybe a solenoid???) the magnetic field within the wire changes polarity in order to repel the north pole. This is so that work has to be done against the magnetic field in order for that work done to be converted into electrical energy.
When the magnet pulls away from the coil (so the south pole is the closest face towards the coil), the polarity of the magnetic field within the coil of wire changes so that the south pole is attracted. Thus work has to be done again to pull the magnet out of the field and thus the work done is converted into electrical energy so an emf is induced.

What I don’t understand about my own explanation is that how can a coil have a magnetic field unless it is a solenoid? And if the coil is a solenoid, then it must already have an emf going through it because a current-carrying coil is used to produce a magnetic field and I don’t know….

I know Faraday’s law explains the magnitude of the emf induced but I’m trying to make sense of the negative (contributed by Lenz) in: induced emf is directly proportional to (-) rate of change of magnetic flux linkage.

I don’t even know if I’m explaining my confusion correctly…