Your original post made me curious, so I put a scope on a small relay that has a 1N4001 freewheel diode.
The relay takes 80 mA at 5V
I found that the coil voltage rose at 6v / microsecond when the transistor turned off. I think that this means that the capacitance is about 13 nF, which I guess is from the interwinding capacitance of the relay
The forward voltage on the diode peaked at about 1.8V for 500 ns before settling down to the forward drop of about 0.7V which it continued for about 10ms until the coil current had stopped.
I think that the peak voltage is because the diode takes time to turn on, but the diode is certainly keeping the coil voltage far below what would damage the transistor.
As I understand the capacitor method of keeping the voltage withing limits, a capacitor is in parallel with the coil. There is no diode with this method. When the current is turned off , the energy from the coil ends up in the capacitor, so the bigger the capacitor, the smaller the voltage surge. The circuit will oscillate, but decay quickly and the peak voltage is well defined.
A resistor in series with the capacitor will only increase the voltage a bit, and it will stop there being a very large current pulse as the capacitor is charged when the current it turned on again.