The correct and normal term for this is deviation and in this case the deviation is 20kHz not 10Khz.
The example with center frequency 100mhz with a frequency shift from 99.99-100.01mhz give 10khz deviation and 20khz swing, swing is the total frequency range covered by the FM-transmitter.
Using 100mhz as center, 2khz modulation frequency and a deviation of 10khz will result in a modulation index of: Mod.index = deviation from carrier (khz)/Modulated audio frequency (khz) =10/2=mod.index 5.
deviation depens on the modulation index we chose
Not fully right, deviation is proportional to the amplitude of the modulating signal. Think of this, if a symetrical audio is fed to a resonant circuit where a varicap is used, then with a certain audio amplitude the result will be equal deviation on each side of the center frequency as long as the amlitude is inside the linear region of the varicap, higher amplitude will make the deviation bigger.
Higher index will increase amplitude of the sidebands and ofcourse increase deviation somewhat because of higher content in the harmonics.
Let me hear what you think guys ;D