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.
Almost two years later, I realized that I never came to an answer here. The kit that Ring provided consisted of angle wedges to turn the doorbell to better focus on the area in front of the door, which did not help me, as it even more blocked the door. What I wound up doing is drilling an additional hole into the front of the post to run wires through, bought some more bell wire, and spliced additional length to the front, where I mounted the doorbell. My primary method string the wire involved using a more stiff wire to initially poke from the side to the front, and then attaching the bell wire to that to pull it through. The only issue I've faced is that I had to seal off the hole to the side to prevent humidity from affecting the bell's operation.
Best Answer
sounds like it's not wired correctly or it uses too much power. When the button is pressed, the line voltage drops, which is why the light on the button goes out. electrically, there are conductors and insulators, and a button switches between the two, whereas your ring is somewhere between a conductor and insulator (a load). Yopu old-fashioned bell is confused by this new electrical gray-area.
If your wiring is correct, and it's still "stuck", you have a few options.
a bigger doorbell transformer. Many are very weak, especally ones made after real buzzers and bells were phased out by chimes and speakers (80s?). Make sure the transformer has enough "watts", "VA", or "ma" (labels vary) to power the Ring.
thicker doorbell wire: lower guage wire drops fewer volts over distance, so it might make up the difference to the ringer enough to avoid the low-voltage triggering state continuously.
disconnect the ringer and rely on your phone to notify you of visitors.