[identity] [shared] Separate identity search messages into client and server messages

Summary:
This separates client and server messages into two different enums for organization purposes. This will be reflected in the client javascript flow types for proper runtime checking.

Depends on D10981

Test Plan: cargo build

Reviewers: varun, bartek

Reviewed By: varun

Subscribers: ashoat, tomek

Differential Revision: https://phab.comm.dev/D10945