True, that is the definition of vegetarian every where you look. And if you read further, it also includes "types of vegetarians..." Not only in the dictionaries, other online resources
"What are the different types of vegetarians?
* 1) Lacto-Ovo Vegetarian: does not eat meat, fish or fowl. Eats dairy and egg products.
* 2) Ovo Vegetarian: does not eat meat, fish, fowl or dairy products. Eats egg products.
* 3) Lacto Vegetarian: does not eat meat, fish, fowl or eggs. Eats dairy products.
* 4) Vegan: does not eat any animal products including meat, fish, fowl, eggs, dairy, honey, etc. Most vegans do not use any animal products such as silk, leather, wool, etc. as well."
Vegetarian Recource Group
There are many different practices of vegetarianism. The following table summarizes the practices of various different types of vegetarian diet:
Foods allowed in the main vegetarian diets"
"Often broken down further into OVO-LACTO, and LACTO. Vegetarians may or may not try and minimize their non food use of animals like vegans."
International Vegetarian Union (IVU)
You can find the same examples of being a vegetarian listed all over the place.