Entered as daily receipts?

Dear

I have two questions:

1st question:

I teach yoga classes that are paid for with vouchers that people purchase from the city (so I don’t receive any money from those vouchers at the time of the class itself).

Am I good in acknowledging that I do not have to enter the people who paid for the class with vouchers as daily receipts?

That invoice for three months = my income, right?

Second question:

When drawing up that invoice, should I designate the city as a belgian company or as a private person?

Thank you in advance for clarifying this here for me.

Kind regards,

Anja