Wiring Fresh Air Damper to New Ecobee Thermostat
Your damper is a 1 wire setup

Basically, your damper is set up as a "1 wire" setup, as that nomenclature assumes that the C wire is returning directly to the air handler transformer, whereas in your configuration, the C wire is returning back to the thermostat first, and then going from there to the air handler transformer.  So, simply proceed using the "1 wire" instructions, making sure that both C wires land on the C terminal of the Ecobee.