7.65 x 54 is also seen as 7.65 x 53. It is 7.65 x 53.5. Also called Argentine, and some call it Belgian in character.
7.62 x 54r is Russian. I suspect 7.62 x 54 is someones short hand. The "r" is for rimmed, the Argentine is not rimmed. ("r" is also seen after the dates on on Russian items frequently. It stands for "Goda", or more simply, "The year of our Lord".
So without seeing where you are seeing these different notations, it is hard to guess what someone is trying to get accross to you.
(I did note that both of what you put down originally in your post were both 7.62 numbers. I work a lot of gun shows, and the two types of ammo get wrongly requested fairly frequently).
If you have a Russian M-44, it shoots 7.62x54 r (rimmed). The cartridges you want to use must have a rim on the bottom of the cartridge otherwise you cannot use them. Do not attempt to use any other type of cartridge in your rifle, unless of course you have a death wish. Also, if it is military surplus 7.62x54 r, it will have corrosive primers and you must clean accordingly and right away. Shermysan