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There are a number of ways that installment loans count, but the one we talk about the most is "installment utilization" (for lack of a better name). That's when you add up the total of all your installment balances and divide them by the total of the original loan amounts. My guess is that closing the loan that was almost paid off caused your installment utilization to go up.
I don't believe that the FICO 8 classic scores differentiate an auto loan from an unsecured installment loan. However, some of my auto scores specifically complain about my lack of an auto loan, which indicates that auto loans vs. other loans may come into play on those scores.
No, they are coded as installment loans and are lumped together.
You score dropped most likely because there was a low util loan on your file, and now it's paid off. Your lowest util remaining loan is somewhat higher, across scoring thresholds, and therefore you now have a higher aggregate balance.
JMO.