Does the information typed into the “order notes on the cart page” field persist even for payment methods such as Google Pay or PayPal? If yes, which mechanism is employed to submit this information?
I recently saw someone mention that using onblur for custom cart attribute fields (thus employing the same mechanism the order notes use) will store the values even for alternative payment methods.
These features have two different functions. Cart notes are only applicable for orders, and it will still show no matter what payment method you are going to use because this feature is embedded on your theme settings or code, and payment methods are only for checkout and order payment. So, they are two separate fields.
Therefore, when a customer placed an order, regardless of payment method that they used, all their order details including any notes, will be included.