Rename BackgroundTask and attempt ID fields to camelCase across background-agent consumers while moving BackgroundManager construction to a single config object.
Ultraworked with [Sisyphus](https://github.com/code-yeongyu/oh-my-openagent)
Co-authored-by: Sisyphus <clio-agent@sisyphuslabs.ai>
The extracted handleSessionIdleBackgroundEvent was never imported by
manager.ts — dead code from incomplete refactoring (d53bcfbc). Replace
the inline session.idle handler (58 LOC) with a call to the extracted
function, remove unused MIN_IDLE_TIME_MS import, and add 13 unit tests
covering all edge cases.
YeonGyu-Kim