Wallet on my iPhone 13 only has the choice of; "Debit or Credit Card" or "Transit Card". I can't seem to add Loyalty cards. How do I do that?

See the following from Add and use passes in Wallet on iPhone - Apple Support

Add a pass

You may be asked to add a pass within an app after you perform an action such as purchasing a ticket. Or, you can tap Add to Apple Wallet when you see it from the following:

• Wallet-enabled apps
• Mail or Messages
• Web browser such as Safari
• AirDrop sharing
• Wallet notification after you use Apple Pay at a supported merchant
• QR code or barcode
• To scan the code, open the Camera app , then position iPhone so that the code appears on the screen.

Also see the following for other types of cards: Keep cards and passes in Wallet on iPhone - Apple Support