How is an EMF induced in a coil? Is this a correct explanation of induced emf and Lenz's Law?

So I know that emf is directly proportional to the rate of change of magnetic flux linkage.

And Lenz’s law is that the emf opposes the change producing it.

I am assuming the coil already has a magnetic flux (even though I don’t know why but it just seems to make sense).

When the magnet is moved towards the coil, the coil’s magnetic flux repels the magnet in order for work to be done against the magnetic field. As the magnet moves closer to the coil, the work done is transferred into electrical energy; thus producing an emf in the coil.

When the magnet is in the coil, the resultant magnetic flux from the coil and magnet are 0, and so the emf when the magnet is in the coil is 0.

When the magnet moves away from the coil, the coil’s magnetic flux attracts the magnet in order for work to be done again against the magnetic field. As the magnet moves further away from the coil’s magnetic field, work is done against the magnetic field and so this work done is converted into electrical energy once again, producing an emf.

I think this is correct but I don’t understand why the coil has a magnetic field in the first place. I.e this is was the case, wouldn’t a chair repel a magnet if it comes close to it and an emf would be induced in the chair?

Oh, I think I might understand whilst writing this. As the coil is a conductor, it has a high number density (a lot of delocalised electrons) and so this when the magnet is close to it, the magnet causes the coil to be magnetised? Then we the magnet moves away, the coil becomes demagnetised and so the coil no longer has a magnetic field?

Thanks for any help, I am just trying to make sense of it all.