"Vegetarianism" does not come through the mail with a pocket-sized book full of rules. Though there are many who have their ideals, it would be wrong to criticize any vegetarian for they too, no matter how small, contribute to the environment.
Now, it depends on a person's reasons for becoming vegetarian that decide what type of vegetarian they are. Some vegetarians eat fish, or even chicken. In my mind, however, any person who refuses to consume some form of meat is one less supporter of a terrible industry that proves only to ruin the health of humans, end the lives of innocent animals, and destroy the environment for all living beings.
My dietary vegetarianism breaks down like this: I do not eat any meat, this includes pork, beef, turkey, fish (of any kind) or chicken. I will not eat soups that contain meat products or by-products. I do drink hot chocolates made with cow-milk, but try my very best to always drink non-dairy milk at home (oat milk is my favorite, but I also drink multi-grain milk). I do not eat eggs scrambled, fried or otherwise prepared. However, when in restaurants, I will eat pancakes or cookies. Not french toast. When I cook, I use an egg replacer and also do not use milk in my recipes. (I make great vegan chocolate chip cookies [made with vegan chocolate chips, for the curious].) There are times when no matter what type of food it is, I will decide that if it has animal products/ingredients, I will not eat it. I no longer buy leather products, though I haven't decided as firmly about wool.
I am not a vegan, but do consider myself a strict-vegetarian.