Make password manager authentication one exclusive choice

The password manager's settings menu had three checkboxes whose
interaction was unclear: "Require Authentication to Open" only asked once
per launch, which users took to mean every open (#13084). Replace them
with four mutually exclusive choices: never, once per launch, once per
launch and after screen locks, and every open. The stored settings are
unchanged.

Every Open now also closes an open password manager window when the
screen locks.

Secure setting batch writes go through a shared helper that skips
unchanged values and rolls back on partial failure. Failures show the
save-failed warning, except when the user cancels the admin prompt.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>