[STG-634] add user metadata into session create

[Kylejeong2]

what

Adding user metadata into session create so we can track which stagehand sessions were created by MCP.

[greptile-apps]

Greptile Summary

This PR adds session tracking metadata to the Browserbase MCP server by introducing a hardcoded userMetadata object with mcp: true to all sessions created through the StagehandStore. The change is implemented in src/stagehandStore.ts where the userMetadata field is added to the browserbaseSessionCreateParams object passed to the Stagehand constructor.

This modification enables the Browserbase platform to distinguish between sessions created through the MCP server versus other Stagehand integrations or direct API usage. The metadata gets attached to every session at creation time, providing valuable analytics and debugging capabilities for understanding usage patterns specific to MCP server sessions.

The implementation fits naturally into the existing session creation flow in the StagehandStore class, which already manages session instances and their configuration parameters. The change is non-breaking and transparent to end users, as it only adds internal tracking metadata without affecting the public API or session functionality.

Confidence score: 5/5

• This PR is extremely safe to merge with virtually no risk of production issues.
• The change is minimal, well-contained, and adds only tracking metadata without modifying core functionality.
• No files need additional attention - the single change is straightforward and self-contained.

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