An invoice is what is owed, and a transaction is money moving at the payment provider; the order holds no money itself. A purchase makes both. An invoice starts as a draft, is issued, then paid, voided or refunded, and only a draft can be deleted. Transactions cannot be written through the API: they record what the payment calls did.
Payment methods are set up in the admin and only read here; the list is public, so a checkout can show it before anyone signs in. The routes the payment providers and the PayPal script call, the webhook and the PayPal order routes, are left out.