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Dakota44:
Too many times instructions are given on very specific items like wire numbers, wire colors, terminal numbers or letters, and what connects to what. And these instructions fail to describe the basics of what is being done.
I will start with your first drawing. I have a hard time reading it.
Your original power line thermostat is probably nothing more than a thermally operated SPST switch that closes when room temperature drops below a set-point and does not open until temperature rises above another set-point.
Almost certainly the two set-points are not independent, but are mechanically related, and only one adjustment exists to control both values. The difference between on and off is called hysteresis. There is possibly another independent adjustment that that allows adjustment of the hysteresis.
This thermostat will have only two switch terminals, and thus, two wires to whatever is controlled. Since it is a line voltage thermostat, but it does not require line voltage, it probably can switch higher currents, and tolerate higher voltages than a 24 V thermostat.
It is unlikely the thermostat requires any external or internal power. All power to operate the switch is derived from the thermal power from the variation in room temperature.
You could replace this thermostat with a simple toggle switch, and have a human flip the switch on and off as required. Like putting a log on a fire when more heat was needed.
What is a relay? In the old days in electrical terms it was an electromagnet (isolated from anything else) that magnetically interacted with an armature that in turn could change the state of electrical contacts. Many of these contacts might be isolated from one another, but some might be arranged as a SPDT type of switch.
In general a relay can be considered to be a power amplifier. A small input power can control a large output power.
A 24 volt thermostat is essentially the same thing as a line voltage thermostat except that it is not designed for as high a voltage, and only good for a much lower current.
To use the 24 volt thermostat in a power application we need the thermostat (it is only a switch contact), a relay (has an output switch contact), and a power source for the relay. For very good reasons we need an isolated contact on the relay to control the load.
Your new 24 volt thermostat is basically a SPST switch with two output terminals. Probably closes on low temperature. Thus, only two wires are required from the thermostat to the relay. Likely no power wires to the thermostat, but in an electronic thermostat there may be batteries.
The thing you call a relay is more than a relay, and I don't really know what it is, but I will assume it is simple. By simple I assume it is a 120 to 24 volt transformer, and a 24 volt AC relay coil with a DPDT output contact arrangement.
Looking at the wiring diagram you posted at #9 and the left diagram we see internally what appears to be the output of a DPDT relay.
Terminals N and H at the left are 120 V power to this device. Power must be maintained at all times. N probably means Neutral and H Hot. I don't know that it matters which is which, but use as indicated.
Forget all the other wire connections. All you need is one relay output contact.
Let's consider safety. If you were to loose N to H power you probably would want your relay output contact to open. Suppose you use the left set of contacts with terminals labeled 3, 4, 4 (an absolutely stupid labeling), then I am guessing that 3 and the left 4 are Normally Open with the NO designation meaning that with no power to NH that this contact is open. Do a test on your relay with no power applied and see if 3 to left 4 shows an open circuit. At the same time check that 3 to right 4 shows a short circuit.
The only other question is what the boiler and circulator do or should do when the contacts open from your present line voltage thermostat. If failsafe is in the direction you want, then you want contacts from your relay that open with loss of power to NH.
You do not use any wiring shown in the diagram other than power to N and H, and isolated connections to one output contact.
Wiring from the low voltage thermostat has to be done to get the correct phasing.
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