What it means
Total Cookie Protection isolates third-party cookie storage by the first-party site so a third party cannot reuse one identifier across many sites. Clearing a site's cookies deletes stored state for that site, often signing you out. Neither action guarantees a broken login will be fixed. Mozilla notes that a sign-in loop can arise from third-party storage assumptions but may also be caused by extensions, service outages or other settings.
A real-world example
An embedded login on one site loops under Firefox's cookie partitioning. Clearing that site's cookies signs the user out but does not change the third-party flow. A controlled shield test helps identify whether tracking protection is relevant.
What to do
Reproduce the error, test a narrow site exception, restore protections if it does not help, and clear only relevant site data if evidence points there. Report a reproducible compatibility failure to the site or Mozilla.
The distinction that matters
Total Cookie Protection reduces cross-site tracking rather than deleting your browsing history. Clearing cookies removes stored data but does not prevent new tracking afterward. Global cookie changes can have larger privacy and usability effects than a verified site-specific repair. A site may work in another browser simply because its extension set or login state differs, so that comparison alone does not prove Firefox cookie partitioning is the cause.
Mozilla cookie-partitioning FAQ