It's official: Eggs and dairy are NOT animal byproducts (source: Taco Bell)

There is a difference between vegetarian and vegan. Hence there being two different words.

They're also not animal byproducts according to the European Food Safety Authority. They're instead animal products.

TIL

Animal by-products, as defined by the USDA, are products harvested or manufactured from livestock other than muscle meat.[3] In the EU, animal by-products (ABPs) are defined somewhat more broadly, as materials from animals that people do not consume.[4] Thus, chicken eggs for human consumption are considered by-products in the US but not France; whereas eggs destined for animal feed are classified as animal by-products in both countries

So USDA disagrees with Taco Bell, but the various badges given by the vegetarian association have more precise definitions. I didn't know about animal genes used in plants?