In your account go to the Admin function (gears at the top right) and choose on the left hand side Sales-> settings.
On this page you can enable the option Show treatment date per treatment on invoice and save the settings:
When you pay for an appointment from the agenda, the treatment date will also appear on the invoice.
I have an appointment on 9 May that was settled in the agenda on 2 May.
You can see both dates on the invoice:
The treatment date will only appear on the invoice if there is an appointment attached to the invoice.
So, not if you pay from the cash register without selecting an appointment.