Hi! Don't judge me harshly if you don't understand some part of the message (English is not my native language). I'll go straight to the question: "There was a version of RC2, as soon as it was closed, RS3 came out literally a month later; were the player characters with their levels and personal belongings transferred from RS2 to RS3?"
I believe that Runescape turned into RS2 thanks to EoC (Evolution of Combat) which in turn turned into RS3, your information will have been kept and your skills and items should be unchanged.
Old School RuneScape (OSRS) however, was a re-release of the old game which caused players to have to start fresh again (from level 3 tutorial island). If you have lost any skills/items in RS3 it may be worth contacting Jagex.
As pointed out already, the progression of the game was a continuation just renamed as we moved along. For backstory as to how the game got called RS2 to begin with... Originally it was just Runescape then Jagex decided to put through changes and elevated the game. At the time Jagex gave players an option to have all their items/stats progress in the game or keep them on the original version which got renamed to classic. Thus the era of RS2 began.
When Jagex introduced evolution of combat many players were unhappy with the new combat system. So Jagex put out a poll about it and found a full game save copy from 2007 and created old school runescape in 2012. A new version of the game based off of the end of 2007. When Jagex put through the graphical overhaul to the main game they repackaged the game and called it Runescape 3.
So yes they're the same game that we all played many moons ago just with a new name.Reaping another soul from the ambient mortality