BDP Thero should be Part # HH07QW001

"I do not have a voltmeter and would rather not be fooling around with live wires if I can help it. In the breaker box each zone has two breakers connected with a label "20 amps". Does this help?" And just how do you plan on ensuring the circuit is off before working on it? This is past what you know how to do and are comfortable with. Pay someone to perform the work. Do you have a way of measuring the torque of the screws used for the wiring?

In a two wire thermostat which is controlling heat only, the connections are 24 V power (R) and heat (W). When the thermostat calls for heat, the W terminal is energized to 24 V. You have to trace the red and white wires to seen how they are connected in the boiler. I would think the red wire is the R connection, and the white wire is the W connection, but that is not guaranteed. The terminals in the picture make no sense since nothing is connected to W, and there is a connection to Y which is normally used for cooling. I am not sure the purpose of this connector. Do you have an AC controlled by another thermostat? Does the Honeywell thermostat you purchased have WiFi? If it goes then you need a ground connection. There should be a ground on the furnace control board. Once you find it you will have to run a wire up to the thermostat.

Do you have a gas furnace and AC? When you say furnace comes on, are you saying the blower is running, but the outdoor condenser is not? Where did you connect the wire which was originally connected to Y/Y2? Did you set up the thermostat properly?