No, these two definitions are compatible. Eating an egg is never vegan. Do a search for a video by Earthling Ed on the ethics of raising backyard chickens. He will explain to you that once you commodify an animal, you are no longer a true vegan. True, there are many vegans who slip-up or who are less strict with themselves…but you should always do what is right. Otherwise you are not really a vegan, you are a wannabe vegan.
When i became vegan i assumed there was a ‘1 rule fits all’ but there didnt seem to be. I found this very confusing as i hate grey areas. I literally wanted someone to say ‘dont eat this’ , ‘do eat that’ and it didnt happen. I got on loads of sites and saw vegans with dogs and cats and chickens which confused me also. I have a dog and a cat and when i went vegan i felt at odds as i could never get rid of them even though i guess i am exploiting them. I also rescued chickens (and still do) as i felt it the right thing to do to save them from slaughter.
I guess its about self-justification, i dont care about the definitions as i know i am doing alot more good than harm. I dont eat meat, fish, milk. eggs, butter etc etc and i look after my ‘pets’ very well, the chickens i rescue are ‘pets’ too and thanks to me live anouther 4 years (approx) and are looked after very well too, they even go to the vets.
Ive realised not to get too hung up on definitions, i can do without all meat products and only see that as a positive thing regardless of definitions.
If you feel it wrong to have pets, dont have them. If you feel it wrong to rescue chickens, dont rescue them. Just be ‘your definition of a vegan’ , and remember you are doing more good than harm.