[not_so_insig]

Apologies for the title I couldn't think of a better one. If anyone can suggest a suitable alternative lmk.

Is there a word for someone who doesn't eat eggs and consumes very little cows milk? That eats meat, fish and other dairy products such as yoghurt and cheese? That eats honey and other non vegan products but doesn't necessarily exclude a vegan/vegetarian diet?

I tried Google but it came up with pescetarian but that doesn't seem to fit.