It’s probably neither, or a bit both. Your body and the bell need to do different things for most of the time, so you can’t have a simple answer.
+1
I never think about the swing in the terms of the OP's question (forward and back/up and down). I think about what my body is doing (hinge to plank and plank to hinge) and how it interacts with the bell. When you are learning the swing, do a ton of KB deadlifts to groove the hinge and get a feel for the top and bottom positions of the swing. Once you have the body positions grooved, it's a matter of learning to interact with the bell ballistically -- how to transfer force to the bell to launch it, how to relax to let it float, and how to absorb the force of the bell to load up for the next rep.
This video from Joe Daniels captures quite well what I wanted to say about the bell projection. Although he specifically explains this for snatches and cleans, for me it is the same movement for all three ballistics at this very starting phase.
Keep in mind that the hardstyle would remain planked feet, what his legs are doing is more GS.
He talks about this drill in terms of activating the triceps on the downswing. But it puts me in mind of one of my most used cues in my own practice, "Arm Down!" This is a cue to keep the arm down and connected to the body out of the hole and through the hip extension into the upswing.
He holds his arm down into his body against band tension as he goes into and out of the hole. I use the cue "Arm Down!" out of the hole to get better power transfer from the hips, through the torso and shoulder and down to the bell, as well as to prevent pulling with the arm. It's a little counterintuitive because I am thinking arm DOWN while projecting force into the bell to swing it UP. Since the bell is pulling down, the natural tendency is to think about resisting it -- pulling up. This drill with the band puts the tension in the opposite direction. The band pulls up, so you have to actively keep the arm down.
Keeping the arm down as long as possible, until the hips extend into the plank and momentum carries the arm away from the body into the float, makes a big difference in power transfer. Any time I feel like my power is decreasing and I start thinking about keeping my arm down out of the hole, I get an instant power boost. I find it is equally applicable to swings, cleans and snatches.