[native][web] Align logic in SQLiteDataHandlers across platforms

Summary: Following discussion here and here, I wanted to put up a separate diff to align these two components. Will add inline comments with details.

Test Plan: Check the stamping / DB clearing logic on both platforms by logging in and then logging out

Reviewers: tomek, kamil

Reviewed By: tomek

Differential Revision: https://phab.comm.dev/D11694