[PM-26778] Make VaultTimeoutService use LogoutService

[quexten]

🎟️ Tracking

https://bitwarden.atlassian.net/browse/PM-26778

📔 Objective

Moves logout logic out from vault timeout service and removes it as public API.

📸 Screenshots

⏰ Reminders before review

• Contributor guidelines followed
• All formatters and local linters executed and passed
• Written new unit and / or integration tests where applicable
• Protected functional changes with optionality (feature flags)
• Used internationalization (i18n) for all UI strings
• CI builds passed
• Communicated to DevOps any deployment requirements
• Updated any necessary documentation (Confluence, contributing docs) or informed the documentation team

🦮 Reviewer guidelines

• 👍 (:+1:) or similar for great changes
• 📝 (:memo:) or ℹ️ (:information_source:) for notes or general info
• ❓ (:question:) for questions
• 🤔 (:thinking:) or 💭 (:thought_balloon:) for more open inquiry that's not quite a confirmed issue and could potentially benefit from discussion
• 🎨 (:art:) for suggestions / improvements
• ❌ (:x:) or ⚠️ (:warning:) for more significant problems or concerns needing attention
• 🌱 (:seedling:) or ♻️ (:recycle:) for future improvements or indications of technical debt
• ⛏ (:pick:) for minor or nitpick changes

[github-actions]

Checkmarx One – Scan Summary & Details – c70cb747-5229-49c5-9c1e-d21310f6610f

Great job! No new security vulnerabilities introduced in this pull request

[sonarqubecloud]

Quality Gate passed

Issues
0 New issues
0 Accepted issues

Measures
0 Security Hotspots
0.0% Coverage on New Code
0.0% Duplication on New Code

See analysis details on SonarQube Cloud

[codecov]

Codecov Report

❌ Patch coverage is 62.50000% with 3 lines in your changes missing coverage. Please review.
✅ Project coverage is 38.86%. Comparing base (e9cd6d2) to head (ff143a1).

Files with missing lines	Patch %	Lines
apps/browser/src/background/idle.background.ts	0.00%	2 Missing ⚠️
apps/browser/src/background/main.background.ts	0.00%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main   #16820      +/-   ##
==========================================
- Coverage   38.86%   38.86%   -0.01%     
==========================================
  Files        3420     3420              
  Lines       97312    97311       -1     
  Branches    14625    14623       -2     
==========================================
- Hits        37822    37820       -2     
  Misses      57835    57835              
- Partials     1655     1656       +1     

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[bw-ghapp]

Changes in this PR impact the Autofill experience of the browser client

BIT has tested the core experience with these changes and all feature flags disabled.

✅ Fortunately, these BIT tests have passed! 🎉

[bw-ghapp]

Changes in this PR impact the Autofill experience of the browser client

BIT has tested the core experience with these changes and the feature flag configuration used by vault.bitwarden.com.

✅ Fortunately, these BIT tests have passed! 🎉