fix(web): clarify reconnect session state

[wolfiesch]

What changed

• distinguish a host transport reconnect from a genuinely unreachable host
• show a separate connected-but-syncing explanation while the session inventory catches up
• pass the current desktop connection state into the session composer
• add regression coverage for reconnecting, syncing, cached, offline, and live states

Why

During an expected appserver restart, session freshness briefly becomes cached or offline before the inventory finishes syncing. The composer previously treated that per-session freshness as proof that the host itself was unreachable, even when the desktop transport was already reconnecting or connected.

User impact

The transcript remains readable during recovery, and the composer now accurately says whether Omperator is reconnecting, syncing, or actually offline.

Validation

• pnpm check
• web test suite: 87 files, 1,250 tests
• Playwright E2E: 32 tests
• Android debug build, unit tests, and lint