Most payment means are just that: a way to give money to someone else, and the company/organization/network/system allowing it does not care why you make the payment (other than for Fincen-type reporting, maybe) nor whether you actually got in return what you expected (goods or service).
That's the case with cash, checks, wire transfers, most debit cards, money orders, Western Union, etc.
For those payment means, either there's no way to have the payment reversed at all, or it can only be reversed in case of unauthorised payment.
With those payment means, if you don't get the service/goods you ordered, you'll have to take the issue to the merchant itself, and if you don't get satisfaction, to the courts (usually small claims court).
So, no, a money order will not provide more protection than a debit card (I would actually say less). The Post Office will not refund you if there is any issue with the merchant.
Some other payment means include additional protection, where if there is an issue with the purchase, you can file a dispute or chargeback, they will ask the merchant for some evidence (e.g. that the item has been shipped and delivered), and reverse the transaction if they can't provide the evidence.
This is usually the case for credit cards (usually not debit cards), as well as some platforms such as Paypal.
What cases are actually covered or not vary a lot. There may also be a lot of conditions attached (e.g. Paypal will cover you if you make a purchase, not if you send money directly). In many cases, if the merchant actually shipped something and they have proof of delivery, you won't be covered if what you received isn't what you ordered, or it's faulty, or whatever other issue. Services are quite difficult to get protection for.
Nevertheless, even if you don't get a refund, the payment system provider will usually take a note of the issue, and if there are too many for a given merchant, stop working with the merchant, so they're usually quite attentive to that.