To all who have replied, thank you very, very much for your comments! I tried to respond to each individual post, but for some reason the system is blocking my attempts. So..I will try a general reply to all!
No, the fruit I speak of is not the cherry. Although I did have much satisfaction in sampling fresh, ripe cherries on my visit two years ago! Yum!
The fruit I speak of is the size is.near correct, for cherry and the color is close to yellow or golden cherry. But the skin is more mottled, and the flesh is more the texture of a plum, or an over-ripe pear. Also, the fruit has a blossom end, overall shape being similar to pomegranate.
I think now the correct answer is "loquat", as posted by Elena from Sochi! Thank you, Elena!
As to my purpose, I very much enjoyed this tree, and especially it's delicious fruit, during my visit. I thought how wonderful itt may be to raise and grow this tree for my very own! The climate near my home is very similar to Sochi...even the beautiful Magnolia tree, which lines the streets of Lazarevskoye, is native to my home. And so, I brought many seeds home to try the following year...but I have looked, and looked, and my precious seeds are disappeared! My heart breaks!
Now I have but to find some source for this seed on the internet. My only wish, that it could be seed from "Little Sochi", that I may some day look on it, sample it's gift of tastey fruit, and remember my visit to this beautiful city by the sea!