Reddit mentions of CZH-LABS Electronics-Salon 1PCS Dual SPST-NO 10Amp Power Relay Module, AC/DC 24V Control Voltage.

As I said earlier the solenoid looks very crusty. If water got into that solenoid's coil the corrosion might have shorted the coil out and caused it to draw too much power from your furnace's 40W transformer. This might be what your tech meant when he said it shorted out the board, but then I'd also expect your tech would have replaced the transformer as well or simply replaced the fuse on the board.

To power the humidifier with separate power you would need a relay with a 24VAC (AC, not DC) coil. This particular one is not a good example because it's two relays in one, but for this relay the yellow and blue wire would connect to your furnace control board and then the red and black wires would connect, in series, between a separate 24V transformer and a solenoid. If you change the solenoid you could purchase a 12V DC version, for example, and then power it with any 12V wall wart you have handy; it would be a bit simpler.

The advantage of using the relay is that, in this particular example, the coil is pulling just 15ma from your furnace's 24VAC transformer instead of the 250ma that the solenoid would be drawing if it were directly connected to the furnace. Similarly if you get a short in your solenoid it will only burn up the solenoid or relay and not your furnace's electronics.

Regarding your eBay solenoid, it bears a resemblance to pneumatic solenoids (they're for air only) which you'll find from many sellers. Buyer beware and all that.

Ah, since you have a nest e already, and ordered the transformer I'll make you a wiring diagram that will let it all work together.

You will need to get a 24vac relay and install it on the heater(not hard)

Here is a 24vac relay that's easy to install: Electronics-Salon 1PCS Dual SPST-NO 10Amp Power Relay Module, AC/DC 24V Control Voltage. https://www.amazon.com/dp/B01B60RZPA/ref=cm_sw_r_cp_apa_i_mMsMDbZMQWQDE

Here is how the relay would be wired: https://i.imgur.com/bzXPVIt.jpg

I'll make a full diagram for ya too

If you use an external transformer, connect it across Rc and C. Connect nothing to Rh. Then use 2 isolation relays one for W1 and the second for W2. They make them in dual packages like this one. Electronics-Salon 1PCS Dual SPST-NO 10Amp Power Relay Module, AC/DC 24V Control Voltage. https://www.amazon.com/dp/B01B60RZPA/ref=cm_sw_r_cp_tai_AUHBCb7Z5CZEV . C from transformer to the other side of both coils. These close your W/R control wires for your radiant heat and fireplace.