Thanks: that entry is from Monier-Williams, the premier dictionary of Sanskrit. The entry with abbreviations expanded and interspersing the devnagari script to avoid ambiguities says (ignore the technical term parasmaipadi)
First note this is giving a verbal stem: we need to get a noun and then feminine from that. Not impossible, but ...
I believe the word rai being talked about here is the word meaning wealth (related to Latin res/rem) and not to the one meaning barking. The latter is also an Indoeuropean root, but the verbal form is far more common than the noun form in Sanskrit and it would be pretty peculiar to get from the well known rai verbal stem to a minor noun rai (meaning bark) to riya verbal stem. In other words with a perfectly good रायति, रियति would sound odd. It is far more probable that रियति means to reify. I can't check quickly because it is an obscure enough word that the only citation provided is Patanjali, who is a famous grammarian.
The reference to the verbal stem ri (री) is because that can also give rise to riyati (रियति) exceptionally. That verb means to set free or detach, melt, etc., and is what you see as the meaning dissolved for Rina. I doubt that is the true origin of the name Rina, though.