Ok, lateral bending is a bend through the length of the horse's body, from nose to tail. You want the horse to bend laterally, for example, when you're riding a circle or making a turn. Leg aids can ask for and reguate lateral bend. Generally, a horse that is bending laterally through a turn will stay fairly upright through the turn, as opposed to the "motorcycle" style turn of a less balanced horse. Achieving a good lateral bend in this sense requires that the rider have pretty good equitation: a balanced and independent seat, solid legs, and soft hands.
Picture the curve of the horse's spine being a segment of the circumference of a circle. That is the degree of bend. Small circle = bigger bend.
In NH especially, lateral bend or lateral flexion can refer to picking up on the bit at a standstill and holding pressure until the horse releases his nose, giving to the pressure. I will use this technique on less educated horses to build a foundation of giving to bit pressure and to build a one-rein stop, but after I have a solid "whoa" button, I don't usually flex this way. This type of bending is more commonly (at least in my area) called lateral flexion. It's a tighter, more dramatic looking bend. The vertical version of this is called "breaking at the poll," also done at the standstill, and I'm not crazed about it. Seems to me like it can put horses on their way to traveling behind the bit. Better to teach that frame the "old-fashioned" way, by the training scales.
I think you're okay on the difference between direct and indirect reins. I found this snippet on Wikipedia, maybe it can clarify a little more for you.
Direct rein: one rein pulls straight back, encouraging the horse to turn in the direction of pressure.
Indirect rein: pulls back inward in the direction of the horse's outside hip
, without crossing over the neck. This is usually used to correct straightness problems in the horse's neck and shoulders, as well as for lateral movements
such as haunches-in
Opening rein: does not pull back, but rather the rider moves his or her hands away from the horse's neck in the direction of the turn. This is especially useful if the rider wants to turn in the air when jumping a fence
The full article is called "Riding Aids," there's some good stuff in there.
All photos from Google Search.
Well, that was probably more info than you ever wanted. Hope that was helpful!