Create custom shipping method for a public app

[Blazeorion]

Can I customize the shipping methods that are offered to customers at checkout with the Shipping methods API using a public app or is this only available for a custom app?

I noticed this in the requirements: https://shopify.dev/apps/checkout/scripts/shipping-changes/building-a-shipping-script
You've created a custom app. Shopify Scripts currently only supports custom apps.