No, only L3 and L4 have limited ODD. L5 has no limits on ODD. So a system that can drive anywhere and any time a human can drive is L5. So what Tesla hopes to build is L5.
L3 is defined as full self-driving when the system is on meaning that when the system is on, the driver does not need to pay attention. So when the system is on, the "driver" can take their hands off the wheel, eyes off the road, read a book, do whatever. But if the system encounters a problem, it must give the "driver" enough time to reengage with what is going on and take back control. I don't the think the SAE defines a specific amount of time, it just says enough time. So the system just needs to give a reasonable amount of time for the human to stop what they were doing and reengage with what is going and take back control.
This is one reason why many auto manufacturers, I think, are mostly ignoring L3 and trying to go straight to L4 or L5. The idea is that it is safer to cut out the human completely and just have an autonomous car that can either handle everything itself in its ODD or safely pull over if it can't handle something.
No, what you are describing is exactly L3.
If it can drive anywhere and any time with no geofencing and can handle its own fallback by pulling over to the side of the road like you describe when it encounters a problem, then it is L5.
In basic terms, the levels of autonomy are this:
Level 3: Full self-driving in a limited ODD and human may be asked to take over if needed.
Level 4: Full self-driving in a limited ODD but car does not need to ask the human to take over except when exiting its ODD. Car can safely pull over on its own if it needs to.
Level 5: Full self-driving with no limits on ODD and car does not need to ask the human to take over. Car can safely pull over on its own if it needs to.
Hope that helps.