The standard definition of "vegetarian" in the UK has for many years been someone who follows an ovo-lacto-vegetarian diet. They can (but don't always) eat eggs, milk products and honey, but don't eat animal flesh, including fish.
There are also people who describe themselves as vegetarian, but who eat fish (either with or without specifying that). And there are some people who call themselves "mostly vegetarian" (sometimes omitting the "mostly"), which means they're not vegetarian, because they sometimes eat meat or fish (just like every other omnivore).
When people do things like this it can cause confusion about what "vegetarian" means and what (actual) vegetarians eat, and can make people less inclined to take vegetarians seriously.
So some vegetarians are vocal about the existing, known definition of "vegetarian", and what does and doesn't qualify.
They might point out that there's already a word for "vegetarian but eats fish" (i.e. pescatarian), and it's not vegetarians' problem if pescatarians don't like that term. (I know some argue that people don't know what it means, but there are as many pescatarians as vegetarians in the UK, apparently — if they all started using the word, the general public would soon catch on.)
Or they might say that a "mostly vegetarian" is a type of omnivore rather than a type of vegetarian (or point out that "flexitarian" was coined for these people), so they should call themselves something accurate rather than teaching everyone around them that a "vegetarian" can eat non-veggie when it suits them.
Vegetarians need their word. It's useful to them and everyone around them. Since I'm not one of those people who wants to co-opt the word for my own purposes, I'd say that the people who are most affected — people who eat milk and egg, but don't eat meat or fish — do actually get to say that people who don't fit those criteria are not vegetarians.
So, say that vegetarian groups started to be filled more and more with meat-eaters and fish-eaters insisting they be referred to as vegetarians. And say some actual vegetarians agreed, for whatever reasons, and decided that vegetarianism should be inclusive of those who eat fish, or who only sometimes eat meat.
In those circumstances, why shouldn't those vegetarians who see the value in having a word for themselves advocate for their position? Why shouldn't they get together to campaign just for their own interests as people who follow the ovo-lacto-vegetarian diet, for clear language, for pescatarians to call themselves pescatarians and flexitarians to call themselves flexitarians, and for vegetarian to mean what it has always meant?