The equity payment is simply what you agree to pay the seller for their ownership interest, and it has no fixed relationship to the appraised value minus the loan balance. That gap is the seller's opening ask. What they accept depends on what problem the sale is solving for them.
So the arithmetic that matters is: loan balance you're taking over, plus any arrears (missed payments, late fees, and the lender's reinstatement figure if they're behind), plus whatever you pay for equity, plus closing costs. Those closing costs are real even without a new loan, since you still want a title search, title insurance, an escrow or attorney closing depending on your state, and recording fees. Transfer taxes on the deed also vary a lot by state, and in some places they're a few hundred dollars while elsewhere they're a meaningful percentage of the price.
Where the low rate earns its keep is on a 214k balance at 3.1% versus financing 214k at today's rates. That difference shows up every single month for as long as you hold, and it's why some buyers will pay a seller more for equity than a retail buyer would offer, just spread over years instead of at closing.
One thing to line up early: whoever ends up holding the deed needs a plan for the day the lender notices the transfer and asks for payoff. Refinance or sell are the usual two, and you want to know which one you'd use on this specific house before you wire the seller anything.