Based on your information in the comments:
I simply populated this Support Settings templates, and and used the default. If you have to create an auto-response rule, what are these Support Settings default templates for?
The template configured in the support settings is used when an agent manually creates a case and selects an email to be sent. The definition of this setting on the help article is as:
The template used to notify contacts that their case was created manually by a support agent. The notification is optional; it's triggered by a checkbox on the case edit page. This template must be Available for Use.
The option to use the template that's configured in the support settings is available on new case creation screen in optional section and looks like below.
However, if you would like an email to be sent out immediately after a customer sends an email to create a case, you will need to configure auto response rule. An auto-response rule by definition is:
Send automatic email responses to lead or case submissions based on the record’s attributes. Set up auto-response rules to send quick replies to customers to let them know someone at your company received their inquiry or details about their issue.
While configuring the auto-response rule, you can still use the same template that's configured in your support settings. You can find more on how to configure auto-response rules on the documentation.