[ymichael]

Summary

• Remove activeBackgroundSubagentCount from thread list activity state and fixture/test data.
• Keep sidebar background activity focused on active workflow counts only.
• Update the DB activity query to count only pending local_workflow background task items.
• Clarify materialized background task docs around workflows and backgrounded shell commands.

Validation

• pnpm exec turbo run typecheck --filter=@bb/app --filter=@bb/server --filter=@bb/db --filter=@bb/domain --filter=@bb/server-contract --filter=@bb/agent-runtime
• pnpm exec turbo run test --filter=@bb/db --filter=@bb/agent-runtime --filter=@bb/app
• git diff --check

[ymichael]

Superseded by #303, which folds this cleanup into the combined background work timeline/data-model branch.