The user is redirected to the "API URL" of the template. The parameters are passed by POST method.
The payment system processes the parameters and, if the signature is correct, displays the payment form, processes the payment and communicates with other systems if necessary.
Payment system can redirect the user to the success page after successful payment (if it is spelled out in the settings) or display another notification.
Also, the payment system must send a request to the "Notification URL", a special Tilda server URL. This request also needs to pass parameters and signature by POST method (signature algorithms may differ). Also you need to prescribe the field (and its value) that is responsible for the status of payment in the Advanced Settings. It will be a sign of successful payment.
Only when a request with a correct signature, order number, amount and payment sign is sent to the "URL for notifications", the order will be marked as paid in Tilda.