feat(mosaic): lead UserButton with the active organization, badged with the user

[alexcarpenter]

Description

Preview: https://swingset-git-carp-mosaic-user-button-org.clerkstage.dev/user-button/user-button

Stacked on #10129.

In combined mode with an active organization, UserButton now leads with the organization instead of the user:

• The organization's square avatar leads, with the user's avatar as a small circular badge cut out of its corner.
• The trigger and popover header are titled with the organization's name, with the user's identifier beneath it.
• The trigger's accessible name still names the user, since it is a user button.
• Settings in the header still opens both Organization settings and Profile settings.
• In the avatar-only trigger, the focus ring follows the organization avatar's radius. In the labelled trigger, the avatar is vertically centered and inset from the start edge.
• The trigger's chevron has an equal inline margin on both sides.

Internally, the switcher's member lead is folded into organization, which now carries an optional badging user, and SwitcherAvatar is replaced by a BadgedAvatar that only models the organization + user badge case.

The swingset UserButton page is updated to match.

Checklist

• pnpm test runs as expected.
• pnpm build runs as expected.
• (If applicable) JSDoc comments have been added or updated for any package exports
• (If applicable) Documentation has been updated

Type of change

• 🐛 Bug fix
• 🌟 New feature
• 🔨 Breaking change
• 📖 Refactoring / dependency upgrade / documentation
• other:

[changeset-bot]

🦋 Changeset detected

Latest commit: d5963f5

The changes in this PR will be included in the next version bump.

This PR includes changesets to release 2 packages

Name	Type
@clerk/mosaic	Patch
@clerk/swingset	Patch

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Project	Deployment	Actions	Updated
clerk-js-sandbox	Ready	Preview	Oct 9, 2026 7:25pm UTC
swingset	Ready	Preview	Oct 9, 2026 7:25pm UTC

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