New Provider with a Fee Question

Hi there, I had a question about fee and payment agreements. Just wanted to get some input from someone else in the business.

I originally quoted a parent at a low monthly rate for an infant. She later said she could not afford that but that she could split it in 2 payments and pay me bi weekly instead. So that is what we have been doing. She pays me every other week. Next week will be her on week, meaning she owes me that week. Anyway she asked me if I wanted my payment next week or the week after. Since there are 5 weeks this month. Not sure if you are able to follow this or not. Anyway my understanding is that if you pay me every other week, why would that stop just because there is an extra week this month? Why not just continue what we have been doing?

In our initial meeting, I had written a quote of my monthly rate on the front page, nothing legal, just a note. Just so that she can take it home and think about it and what not. So when she decided she was going to go ahead and go with me, she wrote out on my fee agreement that she would be paying "$$$ bi-weekly." I was ok with that. So now I expect payment every other week not matter what day it falls on. That is what we both signed on.

MAY (what I was expecting this month)
Pay $$$
No Pay
Pay $$$
No Pay
Pay $$$

MAY (what she is expecting this month)
Pay $$$
No Pay
Pay $$$
No Pay
No pay

Does this make sense to anyone, what would you do if you were me?