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Browse credit cards from a variety of issuers to see if there's a better card for you.
There are several scores.
Lenders can use whatever they want.
Most use FICO 8 scores.
CCT report FICO 8 scores.
Credit Check Total scores are legitimate FICO 8 scores-you can count on that. As far as what any particular lender uses, it could one of the many FICO versions, an internal model or (least likely) Vantage 3.
Sounds like you've got some significant battle scars on your reports. Maybe working on getting those cleaned up as well as possible and taking good care of that Cap One card would be a good strategy for the present? Good job on taking a pass on the Credit One offer. Also take a pass on the other predatory offers you are likely to get. Good luck!
Unless you define FICO 08 scores as FAKO, Credit Check Total does not provide FAKO scores.