The term Red Lake is a broad one, and it can sometimes apply to Kermes, though it may refer to the similar crimson made from Cochineal or something else entirely. Kermes, NR3, is a rather rare dying substance which also is made from the bodies of insects but it is distinct from Cochineal, NR4. While both are deep red colorants made from insects, we've heard it takes more Kermes to achieve a deep red dye than Cochineal (the insect from the "new world") does, and Cochineal seems to have superseded Kermes during one era in particular. Kermes can be thought of as a similar insect dye which hails from a different part of the world.
From the perspective of dying textiles, Kermes is praised as lightfast, however art material usage is a different context, and when laked into a pigment there are higher standards, so most red lakes are not considered very lightfast from an archival painting standpoint.
In terms of the disentanglement of historical red lake colorants, the kermes version of carmine is made from a different insect than Cochinael (NR4). Artiscreation says that kermes is the "Laked extract from the bodies of the insect Kermes ilices (also known as Coccus ilicis)," and adds, "It is chemically very similar to carmine (NR3)." Some sources also include the related insect Kermes vermilio.
Kermes is an ancient dye which is mentioned in the Bible and archeology. More from artnet news here on an ancient kermes dye, and more on archaeology here and here.
