In post #3, you posted a diagram. C wasn;t needed in older stats. R and Rc is provided when cooling is fed from a separate 24 VAC transformer. If you have one transformer R and Rc get jumpered. Sometimes they are labeled Rh and Rc which is the better labeling.
R and C get 24 VAC. if you had a battery powered thermostat C would not be needed.
(R & Rc) and C go to a 24 VAC transformer. Think of C as Common.
So a bunch of relays in the furnace/AC use this common terminal and it really is common to a bunch of 24 VAC relays.
R is the other side of the transfomer and it selectively get connected to:
W - Call for heat
Y - call for cool
and G
G - call for fan
Why is there a Y and Y2 on the AC unit? If 2 stages exist. there is also W1, W2 and W3 for the same reason.
The Fan generally may need to be configured.
In heat mode the furnace usually controls the fan. It has to delay onset and make the fan run longer after there is no call for heat.
The thermostat CAN do that role too, but in general, does not.
So, you may need isolated contact closures if the stat needs 24VAC or uses 24VAC to charge the batteries. So, the stat CAN BE "poential free" with a common if battery powered. That COMMON would be R,Rc not C/
To get isolated contacts, you would connect 24VAC relays all connected to the (C) and then a relay to W, another to Y and another to G.
Now you have 3 independent contact closures.
If your missing the 24 VAC because this is a 12V system. 2V VAC is applied between (R,Rc) and C of the thermostat.
If you need the power reduced from the 24 VAC "coils", that can be arranged.
Q