Wiring – ecobee3 thermostat on old 2-wire boiler

Situation:

• 2 lugs on boiler control, one I'll call "red" or "R" that comes from the built-in transformer with ~14v and one I'll call "white" or "W" which, when jumpered to R, turns on the boiler. Very typical old system, except weird voltage (maybe?)

• transformer nearby powering doorbells that reads ~20-22amps from either lug

• old thermostat with 2 wires, red and white.

What I've tried:
– run new wire (I'll call "common" or "C") from doorbell transformer to thermostat location.

• connect R from boiler to Rh on thermostat, W from boiler to W1 on thermostat, C from doorbell transformer to C on thermostat. NO POWER UP of thermostat

• connect R from boiler to Rc on thermostat, W from boiler to W1 on thermostat, C from doorbell transformer to C on thermostat. NO POWER UP of thermostat

I am going to buy a 24v transformer locally and try the same as above swapping it for the doorbell transformer (which is cryptically marked, but may be 16v, since apparently that is a common voltage with doorbells).

My concern is that if the idea is that the 24v transformer is simply serving as the common, the 14v coming from the R lug on the boiler is not enough power to power the ecobee. On the other hand, I could:

• run both wires from the separate transformer to the ecobee call them "R from separate transformer" and "C"

• connect R from separate transformer to Rh on thermostat, connect W to W1 on thermostat, and C from separate transformer to C on thermostat

The concern I have is that the boiler W lug is currently only getting 14v from its own transformer when the thermostat turns on, would upping that to 24v from the external transformer be a problem?

This feels like a weird issue where my boiler switch requires one voltage and the thermostat requires a different voltage. Is it possible that the boiler's internal transformer is just old and weak? That using a new 24v transformer to signal the boiler to start would be ok?

Hope I've explained clearly. I can take and add pictures if needed.

edit: I've sifted through many similar questions, but couldn't figure out if this particular question had already been answered