Alright, here is a timeline of The Ring. It isn’t conclusive, there’s some discrepancies about what ring is what, so this timeline is a comprised of the rings most commonly “agreed” upon as perhaps being the 2012 ring.
Let’s begin with it’s first and only confirmed appearance of the original ring:
13 February 2012 - Stockholm/Paris
Context: Louis had just spent 5 days in France with Eleanor skiing. He returned to England late in the afternoon on the 12th. Harry and Louis were MIA the night of the 12th.
The boys left for promo in Sweden the morning of the 13th. Harry got off the plane in Sweden wearing the ring on his middle right finger.
He then wore it in during promo in Sweden that day.
He wore it to Paris later that day, with fans and out to dinner with the boys and their team:
19-20 April 2013 - Manchester
Context: Harry wore a similar looking ring for 2 days in Manchester during the Take Me Home Tour in 2013. He wore the ring on stage and off on his left index finger. He was not wearing it when he arrived in LA on the 23rd of April which marked the beginning of the Kim Stewart stunt (Louis went to Oman with Eleanor the same week).
27 April 2013 - LAX
Context: Flying back to England for their show in Paris on the 29th, Harry wore a similar ring on his right middle finger. It’s possible this is the Peace ring he still wears as it was first seen the next day in Paris.
It’s interesting that the first time he wore the Peace ring was a week after the last time he wore the similar looking 2012 ring (up until this point).
13 June 2013 - Miami
Context: Harry wore a similar looking ring to the 2012 ring on his left index finger while greeting fans and reporters at the Best Song Ever video shoot.
2 December 2015 - Triple Ho Show Meet & Greet, San Francisco
Context: Harry wore a similar looking ring to the 2012 ring during the meet & greet at the Triple Ho Show radio concert in San Francisco. He wore it on his right ring finger, like he did in February 2012. He did not wear it on stage, but did wear the other rings he had on earlier.
I don’t think that this provides any conclusive evidence, but there it is.