I have two thermostats in my new home. One is an old round Honeywell thermostat. I took off the cover, and here is the wiring I see. I believe this thermostat only controls heat, and not air-conditioning. I have a second thermostat on another level in the house that controls both heat and air conditioning. I went through the help page for identifying my wiring to see if it’s compatible, but I’m still confused. Can anybody look at this and tell me if my wiring is compatible with the nest learning thermostat?
The round is is a barebones 'heat only' thermostat. It is essentially an open/close situation (based on temperature setting) So yes, Nest (or any other thermostat will work fine) - BUT you should provide the Next (or any other smart thermostat) with a 'C-wire'. The C-wire is essentially a neutral wire in the 24 V AC circuit provided by your furnace's control board, to ensure you can provide power to a thermostat. If you don't want to mess with running a C-wire from your furnace's control board, or if it just cannot support one (it happens), external power supplies for that purpose (to power the thermostat) are available as well.
You also said you have a second thermostat. Assuming that one is for a second furnace/AC, as it is not wise to have two different thermostats connected to one furnace. It can 'theoretically' be done but you need to know what you are doing, how and why (i truly see not good reason for 'why').
It does appear that I have one furnace with two thermostats. The main living level has a more modern thermostat, see photo attached, that I would also like to replace with a Nest. That thermostat controls both heat and AC. Am I able to replace both thermostats with nests? Or if not, can I replace the main, living level thermostat with the nest, and leave the old Honeywell in the lower level?
One furnace, one thermostat! Unless your heating system (unlikely) is split up into different zones with its own temperature control.
But yes, you can replace your main level thermostat with a Nest (to control both heating and cooling) - and you should remove the thermostat on the top floor. Better to place a remote sensor up there. Basically, with a system with one heating source/zone, you control the temperature distribution by controlling the airflow. Otherwise, you will enjoy a good temperature where you happen to be, lets say top floor, and down below it will be freezing (assuming you run for heat).
Whether you already have a C-wire there or need to provide one, I have no idea of knowing.