From the description of odd voltage readings, and the fact that you shorted the connection and still couldn't get the bell to ring, I'm guessing the transformer failed. It wouldn't be the door bell button or that wiring since you eliminated that with the shorting. And since you later tested with another power source and verified that the door bell chimes work, that's also eliminated as a possible failure. The only other thing except for the transformer it could be is the wiring from the transformer to your door bell chimes, but that can be a lot more difficult to repair, depending on how far apart the transformer and chimes are. You can double check the wiring by disconnecting it, shorting one side, and checking the ohm reading with your multi-meter on the other side.
You can pick one up relatively inexpensively from the home improvement store, so I'd start there. Make sure to turn the breaker off before touching anything that's connected to the house wiring, and always test the wires for voltage before touching them. I wouldn't bother with repairing the transformer, they're easy and cheap to replace, but hard to repair.
OK, this may be a side-effect of the way the Ring powers itself.
The Ring is in series with the chime. In series circuits, the same amount of current (amps) flows around the loop through both devices. And by the way, this problem crops up with Nest thermostats and dimmers too.
How does the Ring power itself? By deliberately letting a little bit of current flow through the circuit, including through the chime. The Ring (like the Nest and dimmers) is betting that a small enough current won't make the chime (furnace, lamp) activate. And it is guessing at how much current it can get away with.
So my guess is: you have a chime which is more efficient than most. Sure, when the visitor rings the bell, the Ring shorts the two wires to simulate a doorbell push, and then it lets go of that correctly. And then, it needs to recharge its internal battery - transmitting video over WiFi is hard work. So it allows current to flow at that lower rate. Unbeknownst to the Ring, that is enough to ring the chime. The Ring isn't timing out when it stops; it's just finished charging.
Now like I say, smart thermostats have the same problem: they charge by sending some current down the W line, hopefully not enough to engage the furnace relay. If that doesn't work, they give you an option to hook up the "C" line to power the thermostat directly. Unfortunately it does not appear the Ring has an option like that.
TLDR: Your chime is too efficient for the Ring to vampire-charge itself. Get rid of one.
Well, if you're in love with that chime, there might be another option, but it's definitely not UL-approved. You could alter your chime to be less efficient... by putting a resistor of appropriate size in parallel with the chime. Finding the appropriate size would take some measuring or experimentation. You would not want one which gets too hot, worst case someone holds down your doorbell button for a long time. A fellow shouldn't be able to burn your house down by holding the doorbell button.
Best Answer
Replace the chime, the coil is toast from the shorted wires when you weren’t home.
Open circuit voltage with the circuit at rest being 19-20 is nothing to worry about. (a doorbell is a momentary load device, the circuit is at rest 99.99% of the time. I avoided other terms like "open" and "closed" to describe the circut but based on your comment still confused.)