Reddit mentions of LockState Connect LS-24VAC 24 Vac Adaptor for LS-90i

https://www.amazon.com/gp/product/B0071IWJ7G/ref=oh_aui_detailpage_o00_s00?ie=UTF8&psc=1

I have the same set up as you. I bought this above and I just got mine running today! I had already installed 18/5, so I just spliced the wires from the adapter and used the reviews on Amazon with the wire configuration.

So the Ecobee most likely runs off of DC power. This will be supplied by a rectifier and a voltage regulator from the R terminal (24v AC from your HVAC equipment). 24v AC rectifies to about 35v DC. The ecobee will step this down to maybe 12 or 5v DC for use internally. So putting 24v DC on the R terminal should still work as expected, however it's not recommended as it's not a supported configuration. However, it shouldn't cause damage.

To answer your questions, get a 24v AC transformer and relay that accepts a 24v AC signal and capable of switching 1v DC. Your circuit diagram looks correct.

Looks like digikey does not have a wall transformer in stock. In short: the Ecobee needs ~4VA to run. Look for a transformer that can deliver at least 10va (or 10 watts) LockState Connect LS-24VAC 24 Vac Adaptor for LS-90i
https://www.amazon.com/dp/B0071IWJ7G/ref=cm_sw_r_cp_api_hzAtybVP4AJ5X

http://www.digikey.com/product-detail/en/phoenix-contact/2903337/277-10089-ND/4755341

Double check me that these meet requirements.

Thanks for taking the time to explain.
Yes, the Transformer is a wall plug with 24V DC output: https://www.digikey.com/product-detail/en/triad-magnetics/WSU240-0500/237-1459-ND/3094985
The Ecobee is turning on with 24VDC though which confuses me.

So, I'm thinking about ordering this fella which should do the trick, right: https://www.amazon.com/LockState-Connect-LS-24VAC-Adaptor-LS-90i/dp/B0071IWJ7G/

Edit: I would also need a AC relay then I guess, right? Guess I was just assuming DC in the entire circuit. Never did any AC electronics so far...

Edit 2: Just found this quote: "the incoming voltage to the ecobee interface unit is 24V AC (alternating current) but the power going to the thermostat is 12V DC" here https://www.stevejenkins.com/blog/2010/11/product-review-ecobee-smart-thermostat/

Thank you!

So it's looking like I'll be following something like this video : https://www.youtube.com/watch?v=BgDseEF-4OY

After purchasing something like this : http://www.amazon.com/LockState-Connect-LS-24VAC-Adaptor-LS-90i/dp/B0071IWJ7G

Then, I'll plug one wire into C, and the other into Rc!

The thermostat is a Honeywell WiFi Thermostat (RTH6580WF)

The AC adapter was not made by Honeywell. I chose the LockState Connect LS-24VAC 24 Vac Adaptor. Here is a link to the product,
http://www.amazon.com/LockState-Connect-LS-24VAC-Adaptor-LS-90i/dp/B0071IWJ7G?ie=UTF8&psc=1&redirect=true&ref_=oh_aui_detailpage_o02_s00

Thanks for your reply