Feature: Introduce focused Pinia stores

[mjkatgithub]

Goal

Add targeted Pinia stores for auth/session and chat runtime state where shared
reactive state now spans multiple areas.

Scope

• Introduce auth/session store for restore/auth lifecycle
• Introduce chat runtime store for timeline/runtime coordination
• Migrate only high-value shared state from ad-hoc localStorage/composables
• Keep persistence strategy explicit and minimal for MVP

Branch

feature/pinia-auth-chat-stores

Acceptance Criteria

• Auth/session lifecycle state is centralized in a dedicated store
• Chat runtime state has clear ownership in a dedicated store
• Startup and route behavior use store state consistently
• Existing behavior remains stable after migration

Test Checklist

Unit

• Store actions/getters/state transitions for auth/session
• Store actions/getters/state transitions for chat runtime

Integration

• Composables/components consume store state without regressions
• Persistence and restore paths are deterministic

E2E

• Reload/login/chat basic flow works with store-backed state