No, by definition vegetarians eat NO animal flesh of anykind...any animal product that results in the death or harm of an animal. Fish are animals regardless of the sympathy level you have for them. Vegetarians base their diet on grains, legumes(beans), nuts and seeds, fruits and vegetables.
Lacto-Ovo vegetarians also consume dairy and egg products, but no animal flesh of any kind.
Lacto vegatarians consume dairy products but no eggs or animal flesh.
Vegans are the strictest being that they not only DO NOT consuming animal products they also DO NOT use in any manner ANY animal based product of any kind...including but not limited to leather, silk, honey, white sugar(which is processed using bones of an animal).
Meat by definition is the flesh of animals that is used as food. This is typically associated with cattle, hogs and other four footed creatures, but not limited to. BUT this would never by default leave chicken and fish as a vegetable...which it would have to be called in order to be consumed by a vegetarian. So your assumption that all animal flesh is meat is correct and this includes fish.
SO, no, a fish eater is in no way shape or form by any PROPER definition a vegetarian.
My definitions of vegetarian and meat I got directly out of my dictionary.
I hope this helps answer your questions

.