I already know how to change the version of the Lua interpreter by going to the menu-option "Project -> Lua Interpreter".
I see the confirmation by executing a Lua-file containing the line print(_VERSION), which outputs the set version e.g. Lua 5.3.
However, when I type _VERSION into the "Local console" I still get "Lua 5.1".
Any advice on how to change the Lua version of the local console as well?
The Local console uses the same interpreter that runs the IDE itself, so it's not possible to change at this time. There is a ticket open that could enable support for other Lua versions, but it hasn't been implemented yet.
As a simple workaround you can start normal/remote debugging with the selected interpreter (Lua 5.3, for example) and run all the commands you need in that environment (in the Remote Console).