I think you are a project admin, not a Jira admin - in company-managed (classic) projects such as this one, you add issue types by adding them to the "issue type scheme", but only Jira admins can do that, and if you were a Jira admin, you'd have the link to "issue types" just above "issue layout" in the menu you've shown us.
For JSM June Challenge #2, share how your non-technical teams like HR, legal, marketing, finance, and beyond started using Jira Service Management! Tell us: Did they ask to start using it or...
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