I seem to remember the joists running over the tops of the beams?
The first floor joists are drop in joists into the two long sills, with a tying joist in the middle. As mentioned they are dropped into 4x4x3" pockets.
If the flooring is something line 2" tongue and groove you will spread the floor load out over several beams and hopefully reduce the bounce factor. My calcs show a 7/16" deflection in the middle of the floor (not considering spreading the load).
Red pine would be grade #2, size 4x8 at 16" oc.
You can add a center long sill if you wish but it will need to be supported at both ends and probably the middle as well to reduce the bounce.
Good luck with your project.