FHA has MMI (Mutual mortgage insurance)
Fannie/Freddie use 3rd party companies for PMI (Private Mortgage Insurance)
In most cases banks DO NOT own loans nor do they loan their money for 15yrs or 30yrs. They are the servicer of loans. They collect payments etc.
Banks may make their own rules but cannot ignore certain things. Banks originate loans just like mortgage brokers. In most cases 15yr/30yr loans are sold off to FNMA/FREDDIE who then bundle them into mortgage backed securities (MBS) and sold in the bond market.
At 80% LTV based on the value at the time the loan was originated, you have to fight to get MMI or PMI removed. At 78% it should be removed automatically.
There are loan programs for some that allow you to refi with less than 20% equity that don’t require mortgage insurance, but if you currently have mortgage insurance you cannot eliminate it if you don’t have at least 20% equity.