This is my take on turn initiation for carving the groomed.
Initiate the turn with the downhill(outside) ski, with your little toe, by trying to push it down into the snow. That move will push the ski a little forward but more importantly will pull the hips and every thing else along with it into the new turn. If you continue to focus on that little toe into the new turn you will find your weight more evenly distributed on both skies. Then it is a mater of whether you choose to keep it there.
Personally I'm a big fan of the downhill ski, especially on steeper runs. I tend to use my uphill ski a lot in moguls but the dominate ski is still the downhill ski.
The downhill ski will tend to move forward in the initiation and then catch up to, and then pass the new downhill ski through the turn. This is a function of biomechanics, and happens naturally. It can, of course, be reduced or exagerated for various reasons.
I have an issue with that natural move being countered in some of today's instruction methods especially to lower and mid level skiers.
I also like to turn my brain off when free skiing and just feel. I hate it when it gets in the way. I think it is one of those man made obstacles we are warned about on the back of our passes.