Anybody who has purchased ink cartridges know that there is no blue or
red ink. Cyan sure looks blue and Magenta definitely looks red to me so
why don't they just call them blue and red instead of cyan and magenta?

I'll try to clear things up a little. Monitors verses printers use different
ways to produce color. A monitor is a light source while printers are light
reflecting. The only common ground between a monitor and a printer is
what is referred to as dithering. The definition of dithering is as follows:

A process that simulates shades of gray or color variations by differing
sizes and shapes of pixel groups instead of an ordered array of halftone
dots. This reduces the contrast between dots of different colors or shades and yields a more flowing, natural impression.

In other words dithering uses overlapping color dots to make colors and
shades. Depending on how overlapped they are and how the dots are
positioned results in a controlled color output of millions of different color
shades and hues.

Now back to our discussion. Monitors and printers both use primary
colors, however, a monitor uses primary additive colors Red, Green and
Blue while printers use the primary subtractive colors Cyan, Magenta and Yellow. In both cases the primary colors are dithered to form the entire color spectrum. Dithering breaks a color pixel into an array of dots so that each dot is either made up of one of the basic primary colors or intentionally left blank to create white.

Cyan and magenta are actually old printing terms for blue and red but there had to be some way to differentiate them between the additive and subtractive spectrum. So I guess we're stuck with cyan and magenta for now.

So this guy goes to the store and says to the clerk, "I want to buy a red ink cartridge for my printer". The clerk politely responds, "Sorry we only sell magenta." "Bless you", the guy says, "do you need a hankie? Now about that red cartridge". The clerk, nonplussed, repeats, "Sorry we only sell magenta". "You should do something about that head cold", the guy says. By now the clerk is fuming. "Look, do you want to by the magenta or what?" The guy says, "No thanks, but I think I'll get out of here before I catch your cold".