It has always been a concern to me that everything in this series has been founded on a time paradox - Thawne's time-travelling and manipulation alterd time to create a manufactured timeline whih we have had since episode 1. Without that Particle Accelerator accident Thawne engineered just what was The Flash's origin originally...?
Which brings us to the ring... given Thawne was from Barry's future is it not possible, likely even, that the ring originally belonged to Barry Allen, the original Barry?
So what we are seeing unfold is a paradox. Its a shame really as I always liked the Idea Barry was clever enough as a Scientist to create his own Ring in the same way Stan Lee had Peter Parker Invent his web-shooters and fluid. It is possibly the one downside of the television format, that it is an ensemble cast and Barry therfoore lacks his independence and therefore the urge for self-sufficiency.
Well that RF ring is literally an RF ring in that all closeups show its lightning bolt symbol is RF's and not Barry's.
So RF likely copied Barry's but altered it with his own symbol.
Now in the past, Barry shall have the ring examined and replicated with his own symbol so that he will have his own ring to store his costume/clothes.