What you call a stipe is the pedicle, or pedicel of the flower. As you said, leaves have petioles, flowers have pedicles. In bougainvillea, the colorful bracts are also attached to the pedicle.

When you prune any plant, what you want to remove is the growing tip of a branch, removal just the 'stipes' will not have the desired effect. Most plants, when the growing end of a branch is removed, will loose what is called apical dominance. Cells at the growing tip secrete hormones called auxins which prevent the branch from growing side branches. When you remove the growing tip the side branches can now develop, increasing the bushiness of the plant.

Thanks for clearing that up so nicely.

You also want to prune to a leaf node, not the middle of the stem. New growth comes from the node. Look at the node to see which way it is heading, one pointing up, the new growth will go up. But in some cases it will form 2 new branches. Then if you prune the 2, you will get 4, that's why it gets bushier. Hope I explained this well enough.