How the Log Periodic Dipole Antenna of Localiser help in varying modulation depth?
Mod Depth (more commonly called modulation index in electrical engineering) is the amount a carrier signal is modulated.  Mod depth has nothing to do with antenna configuration and is set by the transmitter (or more precisely the modulator).  The log periodic antenna controls the width of the various beams.  A larger antenna means a smaller beam and smaller beams mean you will have more precision on an ILS approach.  As far as the discrepancy between the 40% and 20% mod depth for ILS systems I will always believe the manufacture.  It seems like the documentation you posted is describing a simple hypothetical setup to help you understand the concept.  I believe most ILS systems use more than two beams. The degree of mod depth shouldn't matter too much as long as your Signal to Noise Ratio is high so I wouldn't get caught with the amount of mod depth in ILS systems.