In both scenarios:
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If you're using bash, you also have access to the 'disown' built-in which dissasociates all backgrounded jobs from the current session. That will let you quit your current session and have them continue on. You won't be able to foreground those jobs in another shell so if that's your aim you'll need to look at screen.
The disown command is useful however if you started the job in the background and don't want to have it exit when you log out or close your terminal.
I'm not sure what exactly you're trying to accomplish, but have you tried running the "Screen" command?