CD’s are much more prone to damage from the top. The data layer is just beneath the top surface, and so it is easy to ruin one in this manner.
DVD’s have the data layer sandwiched between two layers of polycarbonate. You would have to make a very deep scratch from the top side into the reflective layer to cause permanent damage. From the sounds of it, you’ve just cut through the ink on the top of the dvd.
I’d probably just leave it alone if the disk is still playing properly. But you can also make a copy and burn it to a blank dvd, or save it on the hard drive if you want a bit of reassurance on keeping this movie.