You might be on to something. I had an old car loan that was paid off still showing on EX and my EX score is 713. My EQ doesn't show the car loan and is 669. Without another car loan or first mortgage would the extra credit card help?
It might not help your score now, but if you get auto or mortgage loans later, it might. Or if you have to carry a balance, you'll have more cards to spread it over.