How many types of credit score are there?

Well, the 3 major credit bureaus in the US which manages and stores credit reports of most of the consumers. So, it depends on your creditors who report your credit usage to these credit bureaus. You can get a copy of your credit report from each of the credit reporting bureaus free of cost once a year. Otherwise, you have to pay charges to get a copy of your credit report.

FICO is the most standard credit score. Most lenders check this score only. So, you should also check this score. Vantage score is also there but FICO is more popular. Citi and Discover allows you to check your credit score for free. Chase and Capital one allows you to check your Vantage score for free.

Well, I would say that you'll get an idea if you check the scores from the 3 major bureaus. You can even do an average yourself. Before opting for al loan, if possible, you can inquire from the creditor/lender which bureau they refer to. Sometimes, the creditor lets you know.
You can request to check your preferred score. They might listen but don't have much hope about it.