Do not under any circumstances connect your relay as they show in your last example from Gravity. Nothing from the Arduino should be connected to the relay contacts which should go ONLY to the high voltage wiring to your boiler.
You need to think about reliability here .
The packaging of what you make , electrical interference , user interface , who maintains it etc .
Much safer and easier to buy a room stat .
If you must put your circuit in parallel with the stat so it has a “fall back “ .
Look at Bogey boards from RF solutions .
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