A.
Changing the ferrite ring by a larger one - 20 mm diameter:
- using 47 uF the LED is brighter than using 10 uF.
- The voltage from which the LED started to light was 0.82 V, while the current was stable at 0.002A (sources shows).
I guess I can confirm then that the value of the capacitor that makes the circuit able to light the white LED is in close relation with the value of the ferrite inductance.
I also believe (but I have no evidence) that some circuits still rise the voltage without a capacitor because the (implicit) capacitance of the circuit is enough to keep the oscillations, at lower values of inductance.
B.
I mounted both ferrite rings (the 13 mm one and the 20 mm one) in parrallel, with the 47 uF capacitor on. The LED started to light at 2.15 V.
For the time being I cant measure the inductance and the vendor has no clue about DS of the ferite rings :-(.
C.
Late edit: once the circuit is on and the LED lights, at about 0.85 V, then decreasing the voltage to 0.56 dims the LED but it still brights.
D.
Later edit: the 1 uF capacitor turns the LED on from 0.56 V (as the source displays).