In my recent riding lesson, I was working on my legs. She told me to almost roll my knees in, but then kind of have my toes out. This felt pretty impossible for me to keep up. Whenever I had my knees set, my calf would come off the horses body. What confused me was that my old instructor told me to relax my knees and not pinch, they would be pretty loose. And with my knees so tight, I feel like I have no weight to put into my heels.

 Hmmm, well, this is the way I teach it:

Relax your whole leg and let it hang nicely by the horse's side.  Think about letting your knee drop and your thigh lengthen, feeling a nice even contact all down your leg.  If you want to roll your leg in, that motion should come from the hip, not the knee, so as to prevent pinching of the knee.  You definitely don't want to feel like you are gripping with your knee or any part of your leg.

That seems to help. She didn't mention the hips. She put a little piece of paper in between my knees and the knee roll and had me ride with it for a little while, and it felt kind of uncomfortable. I feel that my whole inner leg is touching the saddle, but my knee bone doesn't, just the inside of my actual leg where my knee is.

 Yeah, I think she and I may have philosophical differences.  By trying to hold a piece of paper with your knee, you are forcing yourself to pinch with your knee which is exactly what you DON'T want to do when jumping because then your lower leg will swing back and your upper body tip forward, as you've noticed.  

 I wouldn't be able to hold paper with my knees on my horse. Not only do I not ride that way, but do as the other poster described. My thighs are too plump to comfortably get my knees onto the saddle. I focus on keeping my inner thighs and inner calves in soft contact. If you pinch with your knees and the horse spooks or crow hops you are likely to come off rather than go with the horse.

That seems odd to me, like it's promoting a bad habit. If you grip with your knees it will tense your whole body and inhibit your balance and motion. Did she explain why she wants you to do this?

Well, the only thing she told me was it would help me be in better contact with the horse, and it stops him from wobbling so much. 

 I would definitely ask what her rationale is.  Communication with an instructor is really important for the relationship to be helpful.  Perhaps explain to her that you have been thinking and doing research trying to learn more about correct positions and wondered what her thoughts were on how gripping like she wants you to can promote a secure even leg contact and a relaxed seat and leg.  Some kind of non-confrontational question to allow her to elaborate on why she has you doing this exercise.

As several other posters mentioned, I was always taught to grip with my inner thigh and calf, never with my knee for the same reasons above. I have heard of instructors using the hold a piece of paper exercise, but the paper always goes between the thigh and saddle. I've even been to a couple of shows that had game classes(for Halloween) and one of them was to ride bareback with a dollar bill between your thigh and your horse and whoever kept the dollar in place through all of the gaits and manuevers was the winner. I would definitely talk to your instructor and ask her if maybe she's explaining it in a way that you're getting a different message than what she's intending. Let us know what she says, as I'm curious to know her thinking in this exercise.

I have to agree with everyone on communication with your instructor is important.  She may be seeing something that you don't even realize you are doing or not doing. 

Mystery solved.