24 volts is 24 volts . Amp draw from elements does not affect that
I was referring to the VA, volt-amps, or power, of the coil, and whether the transformer had the power to drive it and if the thermostat had appropriate current ratings to handle it. Obviously an 800A contactor has beefier contacts than a 1A relay, so the solenoidal coil that
moves them will be sized to require more power.
That said, the
HVAC world appears to have settled on a cheap contactor design called Definite Purpose. The case style and mounting feet of the DP Compact models appear to be what Dave has. The leads to the coil are down low on opposite sides, with two (connected) male spades on each side. (Which answers one of my questions.) Although DP Compacts only go to a rated current of 40A, that is for motor loads and the beefiest Eaton C25 is rated for 50A into purely resistive loads, which is what the filaments (elements) present. I'm hoping that's what Dave has.
And looking at Eaton's coil characteristics, they're all 8 VA within that family regardless of contact rating and coil voltage. (On the other hand, the beefier "Special Purpose" contactors of similar switching capabilities have 50 VA coils.)
24VAC appears the standard control voltage in residential HVAC so no surprise there. That looks like about a 12W transformer (need to confirm) so we should be ok, though not a lot of margin. And this stuff all seems to fit together well (it would cost a fortune for the average homeowner to service it otherwise) so most home thermostats can probably handle 1/3-amps of control current to the white wire as well.