The current output to the load connected to the amplifier is supplied by the power supply, so it is correct to say the amplifier modulates the current drawn from the power supply flowing in the amplifier load.
However, the link given is pretty wrong in most counts - for example claiming that FET's lowered weight and cost of amplifiers - if anything they might have made them heavier, as slightly larger mains transformers are required to give the same amount of power.