She is more respectful in the other bit because it is more painful, in short. If you can ride her in a mild bit, do it... even if there are a few problems, it's better to smooth them out with schooling than with a harsher bit. If you can't, there are some training issues you need to work with. The only case (in my opinion) where it would be alright to use a stronger bit is because the horse prefers that type of bit over another and therefore works better, happier, and more willingly in it. But riding with a stronger bit for the sake of better control is simply not right, since riding should not be a battle against the horse for control; it should be horse and rider working together.
For now I'd suggest that you stay away from shanks and harsh bits altogether. I take from the way you talk about bits that you are not extremely educated about them, and therefore it's safer and kinder to stay with a simple snaffle. A shank or harsh bit creates the illusion that a horse is more responsive, but really that isn't so. Let me explain: With a snaffle bit, applying 1 pound of pressure on the reins results in 1 pound in the horse's mouth; in the same way, 10 pounds will result in 10 pounds and 5 in 5. With shanks or harsh bits, 1 pound of pressure on the reins could amount to 10 in the horse's mouth. It does not make the horse any more responsive or softer in the mouth, it simply creates the illusion. And for someone who does not have the required experience, these bits could easily do a lot of damage without the rider even knowing.
I highly suggest you begin with a simple snaffle or french link snaffle. If it turns out that your horse prefers a different bit, use that, but don't do it merely because you feel it gives you more control. Start with a snaffle and work out any kinks in the training in that bit.
-----In riding horses, we borrow freedom-----