In addition to writing scripts in Scripting Extension, you can run scripts from Scripting Extension. The other place where you can run scripts is in a Scripting action in an automation.
A script is just plain text. You could store them in a long text field if you wanted, but you could not run a script from a long text field.
For simple scripts, I save them only where they are run -- either in Scripting Extension or the Automation Scripting Action. For more complex scripts, I manually copy/paste to a git repository.
If you want to see the history of different versions of an automation script, the automation history now shows revision history.