Question
from what kind of cactus plant is extracted the color red

Answer
The only way I know that red dye is extracted form cacti is from an infestation of the Cochineal Insect

This pest does not attack all kinds of cacti, but is specialized on the Opuntia genus. The insect family it belongs to is Dactylopiidae and the genus is Dactylopius. There are five species in the U.S.  and are related to mealy bugs. The specie D. coccus Costa has been used in Mexico and still is for the production of carmine red pigments. It has even been mentioned that the red pigments at times are added to the juice of pomegranates to enhance its color. The females are red in body color and up to about 3 mm in length. Once they are mature, they are scraped off the plant and dried. The red pigment is then extracted. This pigment was commercially important until 1875, when aniline dyes were introduced.

On infested plants, typically the entire underside of the stems are covered completely with the waxy-white cotton-like fibers. This material is produced by the insect and helps to protect it as it sucks out the plant juices. While this insect can certainly hinder the growth of its host plant, it seldom seems to actually kill the plant. In the case of collecting the insect for dye it is said that both Mexican women would coat the plant with it and scrape the mature insect off, insuring a continuous supply.

That's the only thing I know about red dye and cactus.  I don't know of any cactus that in and of itself that produces red die. So a pest today was a source of income in the past.  I'm always amazed the people can figure out the benefits.

Hope that helps.  Where di your questions come from?

Maureen