The oscillator is not a tuner, it is a tuneable oscillator. When the radio is shifted to receive a fifferent frequiecy signal, the oscillator frequency is also shifted by its coupling to the signal tuner, to maintain the mixer output at the selected intermediate frequency.
The oscillator's output is a frequency that is always separated from the incoming signal's frequency by the IF frequency. The oscillator and tuner track together.
I think you have enough information about the superhetrodyne principal. AM uses just 1 intermediate frequency, usually 4555 kHz., but as FM is in the UHF band of frequencies, a double conversionis used.The 1st. IF for FM is usually 10.7 mHz.
The basic difference between FM and AM is, in AM the amplitude of the RF signal is varied with the input signal, and the RF signal frequency does not change.
In FM, the amplitude of the RF signal is constant, and the frequency is made to vary up and down by the modulating signal, i.e. the RF signal frequency changes, swings above and below the nominal RF frequency.
The mixer does not inject 2 frequencies, 2 frequencies are injected and mixed to produce sum and difference frequencies, one of which is selected as the output intermediate frequency.
Modulation is the impression of "intelligence" on a RF carrier signal, by AM or FM as above.
To produce a signal of 10.7 mHz for your radio station on 101 mHz., the local oscillator will be running at either 90.3 mHz for low side injection to the mixer, or 111.7 for high side injection.