You are correct, assuming a mortgage means you take over the current loan, including the balance, payment, and rate.
If you are currently on the loan, it is possible to assume and you may want to talk with your attorney about it perhaps, but definitely with the lender. There are issues that may need to be cleared, but sure you could hold onto the rate with the loan. I recently refinanced a home in someone’s name just after the divorce. His ex-spouse obviously had to sign some papers. He most likely could have kept the loan he had , but in his case there was going to be at least a 6 month wait. Each person’s situation may be slightly different.