I spent some time researching this because I couldn't find an answer, but the tl;dr is yes, but not officially.
First: None of these methods use Jagex accounts or servers, and I couldn't find a good way for Rs3 or OSRS. These are all remakes of different eras.
For 2001 / 2003, there's OpenRSC. They have a wiki page for hosting your own server that you can follow: https://rsc.vet/wiki/index.php?title=Running_your_own_server
For 2004, there's 2004scape. You can get a copy of the code and then just run
npm startin a terminal to get it running: https://github.com/2004Scape/Server?tab=readme-ov-file#getting-startedFor 2009, there's 2009scape. Their launcher has a singleplayer option (doesn't require downloading any code): https://2009scape.org/site/game_guide/play.html
For 2012, there's Darkan. You can run this singleplayer and locally, just follow the instructions in the readme: https://github.com/DarkanRS/world-server