I think the confusion arises from the different sorts of Dyes uses. The "Royal purple" was Tyrian Dye, made from the shell of sea snails, known then as Murex, but possibly a host of different species. This was a speciality dye of the city of Tyre, and was praised for it's color fastness and rich and impressive color.

Now, is that the only way to get a red dye? No! One of the easiest means is the cultivation and production of Madder, a relatively easy to grow family of plants. We have archaeological evidence of madder dye from the 3rd century BCE in India, Tutankamen had linen dyed with madder, and Pliny the Elder mentions Madder growing near rome. I believe there are also finds from Pompeii that show the use of madder in roman industry for dye.

It goes to reason that where lower classes wear red, they are not wearing Tyrian Red, but likely some form of madder red. The two are visually distinctive, and I don't think you would easily confuse the two.