feat(avatar): fix badge clipping and align documentation examples

[bae080311]

resolve. #9678

Summary

This PR fixes an issue where the AvatarBadge component was being clipped by the root Avatar component's overflow: hidden style.
It also resolves discrepancies between the documentation examples and the actual component structure.

Changes

Components

• Removed root-level overflow-hidden (where applicable) to allow absolutely positioned badges to render without being clipped.
• Added rounded-full to AvatarImage across all styles (new-york-v4, base, radix) to ensure the image remains circular even when the root container does not clip its children.

Documentation

• Updated AvatarBadge examples in both Base UI and Radix UI documentation to use consistent and recommended classes.

Developer Experience

• Added a fallback for NEXT_PUBLIC_APP_URL in RootLayout to prevent runtime errors during local development when the environment variable is not set.

Verification

• Verified on local development server (http://localhost:4000).
• Ran pnpm registry:build to ensure all registry files were correctly regenerated and formatted.

[vercel]

@bae080311 is attempting to deploy a commit to the shadcn-pro Team on Vercel.

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[APimenta4]

@bae080311 can we please reopen this? would benefit some projects I've been working on

[bae080311]

@bae080311 can we please reopen this? would benefit some projects I've been working on

Reopened — thanks for the interest.