I never really thought much about whether I push up with the outside hand or pull down with the inside hand when turning until my first racing school (w/ Bertil Roos). There an instructor suggested pulling down because it was easier physically/less fatiguing. If I remember correctly one reason given was pulling down works with gravity. This made sense to me and I started to consciously pull down both in race cars, karts and on the street too. Later I read a bunch of different discussions on the topic and found a wide range of theories supporting pushing vs. pulling. When I went to the TDI cup karting qualifier, the lead instructor advocated pushing, and when I further inquired, he said karts or cars, always push because you get better precision/more control.

Here are a few discussions on the topic I can find at the moment:

Regarding the steering wheel - are you a 'puller or a pusher
Hands on the (formula car) Wheel

To summarize, there are supporters of both pulling or pushing that claim each method provides better precision and/or feel, and is easier physically/less fatiguing. Further claims include less body movement, better weight transfer or stiffening the chassis by pushing in karting. More people seem to support the pushing theory, especially in karts, but there are some adamant supporters of pulling too.

Considering there are many respected drivers that support either method, I'm content with thinking it comes down to personal preference. I've since return to just doing what's natural for me, which I believe is something in between.