Load your rented floor buffer with its heavy-abrasion screening disk. Turn on the buffer and run it over the floor forward and back, with the direction of the floorboards. It should take up the top layer of the finish. Focus especially on the damaged area of the surface.
Vacuum up the dust.
Repeat the screening process using the medium-abrasion screening disk, getting the rest of the finish up. The damaged area of the finish should be indistinguishable from the rest of it. Vacuum.
Use the fine-abrasion disk to smooth out the whole surface. It should end up flat and dull, without any shine, but the stain underneath should be the same color as before.
Spread the first coat of polyurethane over the surface with a paint brush, starting at the far end of the room from the entrance. Apply it in a light, even coat, going with the direction of the floor boards. Do the whole floor.
Let the polyurethane set for six hours. Use extra-fine sandpaper to sand it lightly by hand, dulling the new gloss. Vacuum.
Apply two more coats of polyurethane in the same manner, sanding after the second coat but not after the third. Let the third coat set for 24 hours before using the floor.