[FOSSOVERFLOW-25] feat: Support Private Room Bookings (Private Events / Meetings)

Description

Currently, rooms can only be booked by creating a public event. There is no way to book a room for internal purposes like meetings, practice sessions, or informal gatherings.

We need to introduce a "Private Event" (backend terminology) that allows eligible users to book rooms without creating a public-facing event.

On the frontend, this can be shown as something like "Meeting" or "Internal Booking".

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Requirements

Backend

• Introduce a new event type: private_event
• Store this in DB as a private event
• Ensure private events are not visible in public event listings
• Integrate with existing clash detection logic

Authorization

• Only Core Members and above can create private bookings
• Roles should be validated in backend middleware

Frontend

• Add option while booking:
  • "Public Event"
  • "Internal Meeting / Private Booking"
• Hide private events from public calendar/event feed
• Show private bookings only to authorized users (if required)

Booking Rules

• Private bookings must:
  • Use same clash detection logic
  • Respect room availability
  • Follow role permissions

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Acceptance Criteria

• Authorized users can create private room bookings
• Private bookings don’t appear as public events
• Clash detection works for private bookings
• Role-based access is enforced
• UI clearly differentiates between public and private bookings