The point that I did not understand is that they seemed to be identical. The right hand side of the circuit are identical, Uses tank oscillation, with a drive transistor. Also both is influenced by the base of that transistor, which is the input. This is why i am confused. They look the same for me , except for the capacitor C2 in the FM circuit. Does this capacitor forces the amplitude to stay the same, and make the frequency changes ? If I look at the Am circuit, the more voltage on the base, will make the c1 and L1 oscillation decrease in its amplitude.
If a cheap little FM transmitter like this has some AM it doesn't matter that much. The reverse is not true.