The logic for managing cookie persistence files was previously implemented directly within the `gemini-web` client's persistence layer. This approach was not reusable and led to duplicated helper functions.
This commit refactors the cookie persistence mechanism by:
- Renaming the concept from "sidecar" to "snapshot" for clarity.
- Extracting file I/O and path manipulation logic into a new, generic `internal/util/cookie_snapshot.go` file.
- Creating reusable utility functions: `WriteCookieSnapshot`, `TryReadCookieSnapshotInto`, and `RemoveCookieSnapshot`.
- Updating the `gemini-web` persistence code to use these new centralized utility functions.
This change improves code organization, reduces duplication, and makes the cookie snapshot functionality easier to maintain and potentially reuse across other clients.
