@George P.
Hey George -- I'm Katie and I manage customer support at Cozy. Your screenshots didn't come through for me, but hopefully I can still help!
To be clear, your renters do have to sign up for a Cozy account to use Cozy -- that's what accepting the invitation allows them to do.
The first step to collecting payments is setting up a lease term. I hear you about the confusing term, since your lease term in Cozy isn't your legal lease. It's just the window in which you're wanting to collect payment from your tenants. Once you've set up a lease term and invited your tenants to sign up, they then have the option to make payments to you from their account.
You certainly don't have to use Cozy for listing properties for rent or for collecting online applications, but we do offer that option for landlords who want if. If you're just wanting to collect payments through Cozy, you can definitely do that! If you have any other questions or if I can help you get set up, just drop our support team a line and we'll be happy to help.