You've lost me there: the new lower wall section, without a top plate, would be built laying on the floor, then placed under the 2"x8". Fasten the bottom plate to the concrete floor with Hilti nails, and nail
down, through the 2"x8" and into the top end of the short studs used for the lower wall.
Construct the upper wall section, again, on the floor, using both a top and bottom plate. Lift it into position atop the 2"x8", then nail
down again, through the bottom plate and into the 2"x8". You'd need to tie the top of the upper wall section to the main wall of the building, using blocks of wood and nails or screws of the proper length. Are you intending to cover over the window area at the top of the side wall? I purposely built my basement layout room with no windows, as I wanted to have complete control over the lighting, although that's a matter of personal preference. A fully insulated wall, instead of the window area,
will make it easier to control the temperature in your layout area, though.
Have you given any consideration to how you're going to handle the ceiling? That could have a bearing (literally)

on how you construct your walls.
Wayne