"But wait a minute" you say, "the pull of a foil kite is going to be created by the lift generated by the airfoil, right? So they should be the same thing."
Well, you are right, but in the context of power kites we use the terms differently. Kites that are said to have great pull and reduced lift generate their power by the speed with which they fly through the power zone (the area downwind of the pilot, up to about 40-50 degrees from the ground). They are designed to have higher L/D (lift to drag) ratios, which allow them to get much closer to the edge of the wind window and can achieve speeds much higher than that of the local true wind. This also improves upwind performance. When they slow down and fly overhead, they generate less power and thus little vertical lift. Some good examples of strong pulling but low-lift foils are...
A "lifty" kite, on the other hand, will generate power when flying overhead at slower speeds, and that power is felt as vertical pull. Kitesurfers use this lift on their foils to catch air coming off a wave. On land, pilots on land boards often go for airtime when they swing their kite to change direction. Other land riders may use lift for "kite jumping" but this can be dangerous, as the heights achieved can be great and the landings are not always soft or even controlled. All kites can generate lift, given the right wind, so all pilots should be cautious. The Flexifoil Blade 4.9 has earned the nickname "Widowmaker", as it is strong enough to pull an adult off the ground, but small enough to drop you without any floaty "parachute" effect. The Flexifoil Blade kites are the kings of lift and must be treated with respect.
Other lifty kites include...