You are correct with the 2 different types of PMI, one is FHA and the other is an insurance /risk managament company offering insurance based on credit, income and assets along with LTV (loan to value).
FHA PMI is an up front payment, but it also is charge every month. This does not go away for 5 years, while the non FHA PMI can be removed based on that insurer’s appraisal. This type of insurance would be very hard to prove unless you were rehabing a home.
There are other ways to avoid PMI. One would be to get a loan that has LPMI which stands for Lender Paid PMI. This means that the rate you are getting includes a hit for the lender paying for your PMI. Sometimes this makes the monthly mortgage payment lower.
You can also get 2 loans known as a combo loan. This would be a first mortgage and a 2nd home equity loan or a home equity line of credit. These were very common during the days of 100% financing. There is not a single company out there doing those right now.