In the UK and the US, there is a huge variation but I think that the two most common positions are:
Eat milk (and milk products e.g. cheese), eggs, and maybe honey but no other animal products. This is sometimes called lacto-ovo vegetarian or more often just vegetarian. If you see a product or menu item labelled "vegetarian" then it might contain milk or eggs.
Eat no animal products, neither milk nor eggs. This is called vegan.
I see that you are from India where it is very common to eat milk but not eggs. This is not a common position in the UK or US (outside Indian communities) and I don't know any simple name for the position.
This is a problem for many Indians visiting these countries as the label vegetarian is not sufficient to tell that the item is suitable. Vegan items would be suitable but this is unnecessarily strict.
Vegetarians commonly eat honey but vegans don't. I think that opinions vary here.
I have not heard of a vegan objecting to breastfeeding. One important difference between a baby drinking their mother's milk and a person drinking cow's milk is that the mother consented. If we could speak with a cow and she said: "you are welcome to drink my milk", it would be very different. Also, we don't slaughter most baby boys to maximise the number of milk producing girls.
Yes, plants are alive though, as far as we know, not sentient. In fact, it is believed that all known life is related. A carrot is a very, very distant cousin of you and me. You might feel that it immoral to eat them as well but, unfortunately, if you eat no other living thing then you will die. You cannot survive on inorganic items alone.
There is a term "fruitarian" which is even stricter than vegan but it is somewhere between very difficult and impossible to be healthy.
A relatively new position that is becoming common is to eat fish and seafood but not other animals. This is sometimes called pescatarian.
