Somewhat state specific

We bought a four place in July that came with tenants. No leases or rental agreements.

3 were easy, one asked to pay July a week late. We had closed on July 1 so everyone scrambling a bit. Said it was OK this month, but please realize this is a one time grace period.

August, no rent on Aug 1 from that tenant. We texted and she said (variously) I thought I had til the 5th (yes NY state has made it the law that you are not late til 5th - my old lease used to specify late on 3rd. Also, cannot charge more than $50 by law, my old lease was $50 on 3rd, $10 a day thereafter, but I digress). She then also said something like plus, I didn’t pay til the 12th so...

I told her she could certainly pay on the 12th of each month if she wanted as long as she came current on rent. She was confused. Song and dance about difficult times.

I hate excuses. Also mandatory in NYS now is 90 day notice for any tenant who has occupied for longer than 2 yrs.

We gave her 90 day notice. She did pay Aug by the 5th, but I have a really low tolerance for excuses, time wasting etc by tenants. She paid Sep on time. We have no deposit on her.

Obviously, she is very sad. This is a great apartment at below market rent. Pleading etc. We were super specific when we took over - she clearly realized too late a new sheriff was in town.

My question is, what happens if she doesn’t move out on Oct 31? Do we just file a regular eviction? Is there a difference between a holdover and an eviction for cause? What happens if she pays Nov 1 rent )we make them pay electronically)?

I think COVID moratorium expired here Sep 20 (although commercial expires Oct 20)?

Anyone have thoughts?