Language confusion. In the west, "vegetarian" usually means "milk and eggs, but no meat". I understand that in India it normally implies lacto-vegetarianism (ie milk but no eggs or meat).
This wasn't always the case, by the way. Originally, "vegetarian" in the west was usually used to mean what we now call vegan.
Different diets are practiced by all manner of different communities in the US for many different reasons. There are more possible variations than we could ever agree upon terms for.
Some people consider honey non-vegetarian because it comes from animals. Some don't eat the crackers shaped like goldfish. Some will eat dairy, but only if it's known to be processed without certain animal-derived additives. I once met someone who had a simple rule: he didn't eat anything with opaque eyelids.
(Personally, I just aim for 99.9% vegan and I don't worry if some sauce has a tiny bit of fish dust in it.)
I'm just explaining what is actually being communicated in the veg culture (at least in the US). Just about any vegetarian and vegan knows the distinction: vegetarians are known to consume dairy/eggs/honey, vegans do not. If a vegetarian doesn't eat eggs, they will say "I'm a vegetarian, but I don't eat eggs either"
So basically when you read "vegetarian" you can safely assume it's shorthand for "ovo-lacto-vegetarian"
If milk or honey can be considered to be part of a vegetarian diet, then presumably unfertilised eggs can too. Roughly speaking, at least in the UK, if you don't eat dead animals you are vegetarian, and if you avoid all animal products where possible then you are vegan.
Mind you, the only people I know who have eaten human placenta all classify themselves as vegan.
You're using a definition of "vegetarian" that is not standard in the US (and, I'm guessing, Europe). Vegetarian = foods other than animal flesh. Vegan = plant based foods only.
