Fake/Vegetarian eggs are priced the same here, so I made the switch and am only using fake eggs now for most of my cooking.
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Vegetarians don’t eat meat (including fish, although some religious sects have a marketing deal with fishermen to count fish as a vegetable for some reason), do eat animal products.
Vegans don’t eat meat and also don’t eat animal products.
Wikipedia has a nice table: https://en.wikipedia.org/wiki/Vegetarianism#Varieties
But what I'm guessing happened is that the less-strict diet came first (at least in the modern era), so got the "vegetarian" moniker, and then we just needed another work for the strict no-animal-products diet so had to come up with "vegan".
The times I have to explain people "It's just not meat" is through the roof.