I have some general rookie questions
1. Is the hunt/jump seat drastically different? If I want to sometimes to take lesson at the h/j barn just so I can actually get longer time in saddle, would I be utterly confused? She had me sit quite differently. I'm use to a slight forward seat. She asked me to put my hand under my butt and felt my seat bone,she told me the horse would not like that, and had me move my seat a bit back, I felt like i'm not sitting up tall like I was riding h/j, is this right?
LONG ago, when horses were utilitarian, those who rode never took a lesson in their lives. Even 70 years ago people would go over a jump by leaning backwards. (Watch "National Velvet" to see what that looks like.)
The basic "seat" is sitting upright and balanced, shoulder-hip-heels lined up, toes underneath your knee and pulled up to receive the stirrup. Riding fences requires (NOW, the "Italian Method") to ride forward, just like anybody who is racing at a gallop. It's easy to practice riding too forward and you'll fall into the fence if the horse refuses. For H/J you will want to practice 2-point and be able to balance W/T/C in that position, as well as be able to move easily from a deep seat TO 2-point, at will.
2. Leg position, she kept telling me that my legs need to go back more, I felt like it's beyond what that straight line was, or maybe it was must my position not correct in the first place.
3. She said my tush must be squeezed at all time even just sitting. That was very different...
4. Steering. I was taught to look where I was going and pull rein back to hip. The way she taught me was to move rein towards where you're going and the other rein stays in place? Is this a different method?
You steer JUST LIKE when you are driving a car, eyes on the road/arena and looking WHERE you are going, instead of looking down to check your riding position. The reins need to be long enough so that you can easily halt or half halt without leaning MORE forward. Pulling behind you is counterproductive. To rein as a rider you pull back towards you with the inside rein, while giving with the outside rein, rein in between your pinkies and ring fingers. To rein as a DRIVER, you hold the inside rein and give with the outside rein, both reins over your index fingers. The reins should be held ~ 5 inches apart which is the average width of a bit.
My advice is to actively look for some very long trial riding and do it as often as you can the rest of the year. A deep seat is best learned by sitting the horse for multiple consecutive hours--4 or more hours at a time.
Your legs will get tired and you will sit deep in the saddle, type of saddle is your choice. Your legs can't stand pushing you out of the saddle for that amount of time, so your legs will hang naturally underneath you.
It is impossible to squeeze your butt muscles for 4 or more hours. Even professional ballet dancers don't do that.
I think your teacher was trying to teach you all of the cues for a full halt and half-halt, but you squeeze, then release when the horse responds, so I don't concur.