There are in fact two questions in your question.
A bar magnet has generally a permeability different from air. Therefore the response of the coil is changed with a bar magnet inside the solenoid.
the fact that the bar magnet is magnetized or not, doesn't change the response of the coil (except if magnetizing the bar would have changed its permeability). The change of the response of the coil is not due to the constant magnetic field but to the bar permeability, as for any core of any material.
When I tested coils with magnet cores, something surprised me: one might expect the permeability of strong magnets to be high but surprisingly it is very low. The relative permeability of neodymium or Co-Sm is only 1.05!
All common knowledge but without the detail concerning whether or not the applied magnet saturates the core material.
Provided the magnet applied to the ferrous core saturates that core, all of the above is true and the apparent permeability of that core + magnet will be near 1.
If the applied magnet only brings the core to near saturation then you have a solenoid which will appear different to pulses of differing polarity.
This is in line with the fact that the permeability of ferromagnetic materials change with applied magnetic flux, the inductance of a coil with a ferromagnetic core will generally vary with current in the same way it will vary when a magnet is applied to the same core.
Imagine that.... a 1mH coil is only 1mH if the applied frequency and current is just right
"As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality." - Einstein
"What we observe is not nature itself, but nature exposed to our method of questioning." - Werner Heisenberg