Admin bar: replace home dashicon with actual site icon if set

[fushar]

Trac ticket: https://core.trac.wordpress.org/ticket/65088

This PR replaces the home / odometer dashicon, with the actual site icon, if set, in the admin bar. The icon will be 20x20px, with border-radius of 2px and background-color of #f0f0f1.

If the site has not set site icon, then the default dashicons (home / odometer) is NOT changed.

Screenshot

Before

After

Before (mobile)

After (mobile)

Use of AI Tools

I used Codex with GPT-5.5 to help with the CSS implementation, with my supervision.

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This Pull Request is for code review only. Please keep all other discussion in the Trac ticket. Do not merge this Pull Request. See GitHub Pull Requests for Code Review in the Core Handbook for more details.

[github-actions]

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[github-actions]

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[github-actions]

The following accounts have interacted with this PR and/or linked issues. I will continue to update these lists as activity occurs. You can also manually ask me to refresh this list by adding the props-bot label.

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Props fushar.

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