Few questions about Ringdoor 2 install.

I recently purchased a Ringdoor 2 and I have a few questions I need help with. The current place I’m living in is a house with two separate apartments. My upstairs neighbors have their own doorbell and so do I.

The doorbell button we had was the kind that has the two push buttons one of top of the other in the same housing. My father and I replaced the older one with the one that I just took it. The back of this doorbell I replaced has four terminals on the back (two for each bell) but the house only had three wires.

My father and I installed both wires (to that doorbell) to my button and one for the upstairs and this worked, no clue how, all of this is beyond my acknowledge. Any way when installing the ring bell, I wired the upstairs neighbors door bell with their original wire and I’m assuming one that’s the power? And I installed my original wire to one of the terminals but I used one of the wires that came with the ring doorbell and just looped it around what I’m assuming is the power that’s connected to my neighbors doorbell. Like I said, I have zero idea what I’m doing but I’m assuming it would work.

I wired everything up BUT I took the wires off of my chime that’s inside of my apartment. I found out via Reddit that I guess I needed the chime to be wired for the doorbell to be hardwired.

So this morning I connected on of the wires to my chime and the second I put the other wire to the transformer part of the chime, the chime just engaged and wouldn’t stop until I took the wire off.

Does anyone have any idea how I can hardwire the ring doorbell while my neighbors still have a doorbell? Btw my chime is the heath zenith 35 base if that matters.

Any help would greatly be appreciated.