Study this tracing diagram and you will see that once both ends are held you cannot put a twist in the rope.
And based on this tracing experiment I will say that twists do not get in the rope from coiling it [ends held] almost regardless of how you do it. Twists get in the rope when final sum [just before untieing] of the rotations of the belayer and the climber[while they are tied to the rope] does not cancel to zero. We are talking about 360 degree rotations on chimneys ledges and beginings and endings.
For you trad climbers I know it will hard for you to tie each end of the rope to the rope bag.
my experience in the days of using a figure 8 was that they did twist the rope and were not mere just push existing twists to the end of the rope. By drawing a line on a rope as I describe earlier in this post you could keep track of such possible rotations.
One could fix the bottom end of the rope before using the 8. 8's seem to both push existing twists down the rope and create new rotations. But if the end is fixed while rappelling with an 8 no new twists can be added. We need an experiment!
Top Roping: depends on the anchors, extra rope beyond twice the climb height, whether it is twist free to begin with and how many 360 degree turns the belay and climber that do not cancel do while they are tied in. At 2 cliffs where we do a lot of both leading and top roping we use a rope that in length is 2X the climb heights, pairs of ramhorn anchors (modified open cold shuts) and the ropes never seen to get twists. Sometimes when the short ropes are in use one guy uses his 70m rope on these 50 ft climbs and it seems to be a pile of spaghetti.
The butterfly like coil would seem to generate no kinks when the ends are fixed as tied on a rope bag. Collect the rope to the bag by pulling simultaneously or ropes together from the 2 knots. Actually you can lay it in any fashion you want working to the middle marker and laying the middle on the finished top of the pile.
At next use of the rope, if you grab this middle marker and walk it out to where you just begin dragging the rope bag you will have no new twist in the rope. For use pull the rope back to the bag on the belayer's end single stranded and give the other knot end to the lead climber.
To do this you will have to undo any knots and pseudo twists you may have put into it while collecting it. I call these pseudo twists because you cannot put a twist into the rope without rotating one end with respect to the other and since the rope was knotted on both ends the ends could not be rotated.