feat: use external_user_id in embed to save conversation

[AdityaHegde]

We recently added support for external_user_id to scope conversations in embed context. This PR adds support for it in the app by saving the recently open conversation by external_user_id.

Checklist:

• Covered by tests
• Ran it and it works as intended
• Reviewed the diff before requesting a review
• Checked for unhandled edge cases
• Linked the issues it closes
• Checked if the docs need to be updated. If so, create a separate Linear DOCS issue
• Intend to cherry-pick into the release branch
• I'm proud of this work!

[begelundmuller]

Backend looks good

[ericpgreen2]

Don't put externalUserId on RuntimeClient

RuntimeClient is our connection/transport abstraction: host, instanceId, JWT, transport, request queue. After this PR it also carries an embed-only string that nothing on the runtime side reads — it's stored in the constructor and consumed in exactly one place (BrowserStorageConversationSelector) to namespace a sessionStorage key.

Suggest passing the user id directly into the selector instead. In ConversationManager, construct BrowserStorageConversationSelector({ userId: EmbedStore.getInstance()?.externalUserId }). The selector no longer depends on RuntimeClient, and RuntimeClient, RuntimeProvider, and getRuntimeClient revert to their pre-PR shape.

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Developed in collaboration with Claude Code

[ericpgreen2]

Rename the param to signal intent

userId: string | null reads as too generic at the call site — a reader sees new BrowserStorageConversationSelector(EmbedStore.getInstance()?.externalUserId ?? null) and reasonably asks "why does conversation selection know about users?" The selector isn't really using a user id in any general sense; it's using an optional namespace for the sessionStorage key that happens to be filled with the embed external user id today. A name like scopeId (or userScopeId) honors what the parameter actually does without overcommitting to embed semantics.

Add a constructor JSDoc

One sentence explaining when callers should pass it. For example:

/**
 * @param scopeId Optional namespace for the sessionStorage key. Pass when the
 * same browser tab may serve multiple users (e.g. embed with `external_user_id`)
 * so conversation selection doesn't leak between them.
 */
constructor(scopeId: string | null) {

The class-level docstring's "Selection is scoped per project" line should also pick up "...optionally per user."

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Developed in collaboration with Claude Code