The term vegan seems to have been coined within just the last few years, primarily to separate those who are strictly against using or consuming any products at all that have an animal origin, so not only do they not eat eggs or dairy, they don't use or eat anything from an animal origin, including anything produced by an animal, such as honey or wool. There are vegan clothing lines available (shoes, sweaters, etc), meaning that no part of the clothing has any sort of animal based origin - so no silk, no feathers, no fur, and no leather. It also doesn't matter to vegans that in order to care for a sheep properly, it needs to be sheared yearly, and that you're actually doing the sheep a favor by shearing it, and that's the source of wool. But since wool is an animal produced fiber, it's off limits for vegan clothing. [Can you imagine wearing a wool coat several inches thick when the temperatures are in the 90's? That's what a sheep which has not been sheared faces in the heat of the summer. Even angora wool - rabbits molt their entire coat twice yearly anyway. If you pluck the loose fur from a rabbit's coat, it does him no harm. In fact, he is far less likely to develop a digestive blockage from swallowing wads of loose fur, and will be far more comfortable in the process of molting his coat.]

Currently, there is a much wider definition of vegetarian in that it prohibits using animal products that caused harm to the animal - in other words, since no animal had to die to produce eggs and dairy, there's no blanket definition of vegetarian that prohibits the use of eggs and dairy. I don't know how long it's been defined that way, although I do recall back in the late 70's reading about vegetarian diets, and eggs and dairy were considered acceptable for vegetarians back then too. (To clarify, some who consider themselves to be vegetarians do abstain from eggs and/or dairy, but don't call themselves vegan, because of other products they use related to animals that vegans avoid using, such as the aforementioned wool.)

At any rate, I only mention eggs as a source of complete protein, essential fatty acids, and many other essential nutrients acceptable to those who are vegetarians by the general definition that vegetarians avoid animal sourced foods that harmed an animal, but don't necessarily avoid animal sourced foods which don't cause the animal any harm. Whether or not eating eggs causes or worsens problems with arthritis or inflammation in humans was not part of the equation - merely that for those who refuse to eat actual meat, eggs can be a very good source of nutrients, many of which can not be found in plant sourced foods.