add: `with-next-auth-app-router` example

[dylsteck]

Adds examples/with-next-auth-app-router, a clone of the /examples/with-next-auth(now renamed with-next-auth-pages-router) repo that instead uses NextAuth.js 5 and the Next.js App Router. The new example repo features very minimal changes, but might serve as a helpful resource for other developers that want to use this stack in an app router repo.

Motivation

I was personally looking to use this stack for a project and wanted to put the example up in case others find it helpful for their integrations.

Change Summary

• Renames /examples/with-next-auth to /examples/with-next-auth-pages-router
• Creates a clone of the example above at /examples/with-next-auth-app-router, which uses the Next.js app router instead of the pages router and keeps most of the original code

Merge Checklist

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• [x] PR title adheres to the conventional commits standard
• [x] PR has a changeset
• [x] PR has been tagged with a change label(s) (i.e. documentation, feature, bugfix, or chore)
• [x] PR includes documentation if necessary
• [x] All commits have been signed

Additional Context

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[deodad]

thanks @dylsteck!