Fast answer
For most first Martina Franca stays, start in or close to the centro storico if the point is a baroque old-town evening with dinner on foot. Use the Piazza Roma and Palazzo Ducale area when the first walk and civic core matter more than a deep-street room. Move to a countryside masseria only for a car-led trip, use the station or new-town edge only when arrival friction beats atmosphere, and compare Locorotondo before splitting inland nights.
If you only do one thing
Choose the old-town baroque core when Martina Franca is meant to be the evening base, not just an afternoon stop. Choose a countryside masseria only when the car, dinner transport, and late-arrival plan are already part of the trip.