What it means
A site-specific cookie and data removal resets stored state for a selected domain. Clearing all cookies touches many services and can require many new sign-ins. Neither action remotely signs out a stolen device or revokes a server-side session unless the service itself ties sessions to those cookies.
A real-world example
A single shopping site has a broken cart. Removing its local data fixes the cart while the user's email and bank tabs keep their sessions.
What to do
Save unsent work, select the exact site in Firefox's data manager, remove it and test again. Use the service's security page for account-wide session revocation if a device was stolen.
The distinction that matters
A site may store unsynced drafts or offline content, so narrow deletion can still have a cost. Also, related subdomains may have separate entries. Clearing cache only changes cached resources and may leave the problematic cookie untouched.
Mozilla's targeted data controls
