Pay Rent or Quit. What does Quit mean?

[Kyle Meyers]

If my tenant does not pay rent and I serve them with a pay rent or quit notice, what does quit mean? I understand the basic meaning is that if they don't pay the lease is done and I can evict, but what if they choose to pack up and leave before the prescribed number of days in the notice? If they have left, I don't need to evict them, right?

My confusion is whether or not I can still hold them liable for the remainder of the term of the lease or if my pay or quit notice offered them a way out?

Example: I sign a lease for 1 year with a tenant. After 6 months they stop paying, I serve pay or quit, they move. How much can I legally get a judgement against them for? Just the days they were still there between the start of the month and when they moved out? The full 6 months of the term they did not pay for? Some other amount?