The 'C' terminal connection & wire that the Honeywell instructions are talking about provides power for the thermostat's electronic control, while the R and W terminal connections simply connect to the loop in your furnace's circulation pumps.
As with you, my old thermostat only needed 2 wires, and the wiring to it was a harness that only had 3 (three) wires in it, 1 of which wasn't being used. In order to upgrade, I needed to (a) pull new wire through my wall which had 5 wires in total (although only 4 of the 5 are being used with the new installation, 1 pair (R&W) for the circulation pump loop, 1 pair ('C' and 'Rc" for the power from a 24 VAC step-down transformer.) I also purchased a Honeywell AT72D 1006 24 VAC 40 VA, which serves as a persistent power supply for the thermostat, available of the shelf at HomeDepot.
Since I am not using a 'C' wire off of the rely for the circulation pump, but instead I am using the external 24 VAC transformer, It was crucial for me to remove the jumper on the thermostat's terminal connections that had been bridging the 'R' and 'Rc' connections.
Since there are 3 heating zones in my house, the 24 VAC transformer can function as the power source for all 3 thermostats, but I can't comment on the limit of how many thermostats are able to function from the single unit.