While you may see "Generic Shipping" displayed in the UpOrder template builder, we are actually using the email variable code {{shipping_method.title}} which automatically populates the proper shipping method as setup in your Shopify store.
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The "Generic Shipping" shown on our template builder is simply a placeholder within the builder environment. The correct shipping method will be displayed when actual emails are sent out through Shopify.

Please reach out to [email protected] or through the in-app chat if you have any questions!