Payment receipts are sent by whichever payment processor was used for checkout (e.g. PayPal), so they would include whatever details that service normally provides. Note that for PayPal checkouts, the buyer's Transaction ID for sending a payment will not match your seller's Transaction ID for receiving that payment on the same order (weird, but that's just how they do it). It appears PayPal's receipt emails should also include our Invoice ID.
You can also have us send the buyer a thank-you email message, which can include the Invoice ID as well as your (seller's) Transaction ID, along with other details about the buyer's order: