This is all a matter of the frame of reference you choose to view your body motion from. You can select a frame of reference that makes you think in a way that improves your results, of you can select a frame of reference that makes you think in a way that stalls your progression.
If your chosen frame of reference is earth / the mountain / the the snow surface, the hips (effectively your center of mass) NEVER EVER FALL IN towards the inside of the turn. If your hips did "fall in", in absolute motion terms relative to earth, that would mean your center of mass was already moving in the direction that you are trying to turn in. When you make a turn, the whole purpose is to change the direction your center of mass is moving. The reason you are moving your skis out is to STOP YOUR HIPS (your center of mass) from moving out.
Now, if you visualize your frame of reference as your body, and just accept the rest of the world is moving around a coordinate system centered on your person, then you can think of your hips as "falling in". But that is a really poor way to visualize the motion, physics and forces. You are moving through space. Space is not moving around you.
But the far more important reason NOT to think of your hips / center of mass as "falling in" or "falling over into the turn" is psychological. "Falling inward" is SLOW. Go stand next to your pool, and just let your self fall over into the water. It takes FOREVER to get to a highly inclined angle. Falling is just too slow. If you think about "falling" into a turn, that lazy natural sense of acceleration due to gravity as one falls will manifest in your technique in many ways. You will get tall in transition because that gives your somewhere to "fall" from. You will have a slow transition. You will be slow to build edge angles. In fact, if you think about falling in, you will pretty much never be able to build high edge angles. You want to flex/bend through transition and MINIMIZE the rise of your center of mass. If your CoM is already really low, how are you going to "fall in"? You aren't. When your CoM is kept low by flexing through transition, then to start a turn, the only way to get your skis on edge it "reach out" with them. And the lower you stay through transition, the higher edge angles you develop, because you are going to extend your DH leg until it is in that strong position - bent about 20-25 degrees at the knee. Again, the lower you stay through transition by flexing as you come off a turn, the higher your edge angles on your next turn will be, and the quicker your will develop those high angles. The only reason not to stay low in transition is pragmatic. At some point you just get too tired to do it. It is exhausting because you never get a moment to let your leg muscles relax and move blood. Evern WC racers are staying less low at the end of the run than at the start. They just can not do it the whole time. It is HARD.
Watch Ted in this video we have all marveled at:
Ted Ligety GS Slow Motion
Specifically, stop the video right at 1:09. Look how low Ted is. During transition he is not just in the 90 degree "chair" position with his femur square to his tib/fib, he is far beyond that...like 30 degrees past square. Does he fall in? LOL...no. He actually gets TALLER and pushes/extends into the turn. He reaches out with his skis. The kinematics are completely different. This is probably a cherry picking of an extreme example, and it is better to watch the whole video and take in Ted's typical transition, but you will see that he consistently gets SHORTER in transition, and taller into the the turn. There is no way to represent this motion as falling in. It is about staying low, bringing your legs and skis under you with as much leg bend as you are physically capable of doing, and reaching out with them as you get taller into the next turn.
Also, do not confuse taller and higher. When your ski is at 80 degrees, in a turn, you are tall, but not high. With your ski 80+ degrees to the snow, your body axis is probably about 25 degrees to the snow. Say your are normally 6' tall, but in your strong edging position, in a turn, your are 5' tall. Sin(25deg) x 5' tall = your head is 25" off the snow. Now, during transition, you GET shorter, but at the middle, you are 90 degrees to the snow Say you bend your legs to the full 90 degrees textbook low transition...and you are 4' tall. Sin(90Deg) x 4' = 4'. You got shorter, but your head moved up 23" as you went from being radically inclined, to straight up and down. So the CoM, the head, they come up during transition, even though you made your body shorter. But by making your body shorter from flexing, you MINIMIZE how much you come up. Most skiers hold the same strong leg position from turn through transition, so they come up way more during transition. Sin(90Deg)x 5' = 5'. In that case, your head would come up 35", not 23". Do that and you're not going to have high edge angles on the next turn.
Flex/absorb through transition and "reach out" with your skis if you want to crank high performance turns. If you want to continue to struggle with developing real edge angles, keep thinking about leaning into the turn, or falling into the inside of the turn.
) - your feet just moved more than your hips. But to you it probably felt like a fall.
