No vegetarians or -etarians of any sort eat any fish or meat of any kind for any reason at any point (doing so means they cease to be vegetarian). Some do eat the periods of chickens some don't. The ones who eat it either don't quite get why using animals and treating them as property is wrong or are fine with the abuse of millions of chickens for some reason usually because they don't really know about, choose to ignore it or have it ingrained in their head over the years that it is ok.
Here is a handy little chart (feel free to copy and use it just don't alter it):
Pescetarian-a vegetarian fish
Pollotarian- a vegetarian chicken
Flexitarian- a flexible vegetarian (a dancer or maybe a gymnast who is also vegetarian)
Semi-Vegetarian - a crash between a semi trailer truck and a vegetarian. Not a very pretty site. Vegetarians need to watch out when crossing the street or driving.
Lacto-ovo- eats dairy and eggs (veal and chicken periods)
Lacto- eats dairy
Ovo- eats eggs
Strict/Pure Veg (often mis-refered to as veganism)- eats no animal products or by products whatsoever
Raw- eats from 80-100% raw vegetables and fruits, nuts, grain and legume sprouts, seeds, plant oils, sea vegetables, herbs, and fresh juices as in heated to no more than 118 degrees (basically a strict/pure veg diet just made raw)
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Vegan- a philosophy and way of living which seeks to exclude — as far as is possible and practical — all forms of exploitation of, and cruelty to, animals for food, clothing or any other purpose; and by extension, promotes the development and use of animal-free alternatives for the benefit of animals, including humans and the environment.
In dietary terms it denotes the practice of dispensing with all products derived wholly or partly from animals."