Currently the kickClient method only sends a generic "Kicked from game (you may have been playing on another tab)" message for all kick scenarios. We need to differentiate between different kick reasons:

• Kicked by admin - needs specific error message
• Kicked by private lobby creator - introduced in Private Lobbies: Add kick player functionality #1436, needs specific message
• Kicked due to multi-tabbing - current default message
• Default fallback - possibly same message as multi-tabbing message

Current implementation

The method currently sends the same error message regardless of kick reason:

client.ws.send(
  JSON.stringify({
    type: "error",
    error: "Kicked from game (you may have been playing on another tab)",
  } satisfies ServerErrorMessage),
);

Proposed changes

• Add optional reason parameter to kickClient() method
• Send appropriate error message based on kick reason
• Update all call sites to specify kick reason

Question: Should these new error messages be added to the translation system?