The story
Hypothetical Hallie was hypothetically a biologist. (She was a "oh, look, nature" biologist rather than a "ooh, atoms" biologist). She observed the natural world, and lo, she observed that (more or less) animals and plants fell into various species. And that, lo, parts of plants could also be categorised: many plants had similar structures that "came from the same specific part of the flower" and "promulgated seeds" and fulfilled a couple of other criterions. She dubbed these "fruits".
This was extremely useful, as if Hallie wanted to make generalisations about the properties of fruits, she didn't necessarily need to deal with each individually: she could first check that each fruit obeyed each criterion, and then draw large inferences from observing only one criterion, and thence inferring that something was a fruit, and no doubt fulfilled the others as well.
However, this rosy simplistic picture did not last very long at all. Quickly, people came to Hallie and pointed out problems. Bacteria came in different sorts, but they didn't really fulfil the criteria to be called "species". "No problem," said Hallie. "We'll divide them roughly into species, but remember there are fuzzy edge cases where a bacteria doesn't really fit." People pointed to the duck-billed platypus. "It's not exactly a mammal, but what else can we call it?" "No problem," said Hallie, "we'll call it a mammal, but remember that there are a few exceptions to the criteria."
However, Naive Nelly was nowhere near as intelligent as Hallie, and rather idolised her. When she saw what she was doing, she also adopted all the clever new words Hallie used. However, she completely failed to notice that they were an (incredibly useful) thinking aid, not an inherent part of the structure of the universe. In fact, she was so stubborn she would often react violently if people tried to persuade her otherwise!
She did the same thing Hallie did, but with the concepts she was familiar with. For instance, she started adding things to the definition of "fruit" such as "succulent flesh good for eating" and "not a vegetable" and so on. And she was right, this was very useful. But she was wrong, because this usage different from Hallie's and eventually they came into conflict.
Hallie recognised that you could normally reason by saying "This has property X. Therefore it's a fruit. Therefore it's not a vegetable." But that sometimes, something had most but not all of the properties of a fruit, and then what could you do?
Nelly tried stubbornly to ensure that such things were, or were not, in the definition. Hallie tried repeatedly to reason with her. "Look, Nelly," she'd say. "It's just a shortcut. It's nice, but it DOESN'T ALWAYS WORK. When it doesn't work you have to actually turn your brain on and decide FOR YOURSELF whether something is a vegetable. Thinking about fruit DOESN'T HELP YOU."
But poor Nelly had got out of the habit of using her brain, and couldn't. She kept insisting that Hallie pick one, and eventually, poor Hallie, fed up of the argument, said "OK, it's much MORE like a fruit than not. In fact, the only criterion of being a fruit it fails was 'not being a vegetable' which was never my criterion in the first place."
But with other concepts, it was much harder to give Nelly a pat answer however much she insisted, and sometimes Hallie would find that if something fit many of the criteria but not some important ones, she would first answer one way, and then forget, and answer the other way.
And poor Nelly would see this as an inconsistency in Hallie, rather than an inconsistency in herself in insisting that things ALWAYS had to fit into categories!
The moral
Categorising things into "vegetables" and "not vegetables" is a useful cognitive short-cut, not a god-given right. Sometimes it doesn't work and you have to think for your fucking self, so sorry about that.
Some people have a bizarre cognitive flaw that "being disjoint from each other" is the most important feature of "being fruit" and "being vegetables". I do not know why people think that.
I think it's a misuse of generalisation. "The first 350 vegetables I examined were clearly not fruits, therefore I expect all the others won't be" is good reasoning. "Therefore all the others are, and I will refuse to change my mind in the face of the evidence, and ridicule and insult people who say otherwise" is TERRIBLE reasoning.
It's an edge case. It doesn't matter HOW you fix it, so long as you recognise that it's questionable, and that it REALLY DOESN'T MATTER. People who go around saying "oh, actually, technically, it's not a vegetable" have made the correct leap that knowledge is good, and many people are wrong about this, and they ought to be educated. But having achieved -- in their small opinion -- intellectual mastery by designating a tomato to be a fruit, and not a vegetable, they completely fail to think that maybe there was something more to say.
The important point is not "whether a tomato is a fruit or a vegetable" it's "is a tomato exactly one of a fruit and a vegetable?" or even "does a tomato fit neatly into either of these categories without fudging?" (Hint: it fits "botanical fruit" almost perfectly. It fits "culinary fruit" badly. It fits "vegetable" quite well but not completely.)
This sounds irrelevant. Sure, even if people ARE wrong about tomatoes it doesn't matter. But it happens to other words too. Eg. "Is copyright violation theft?" It shares SOME of the characteristics of theft BUT NOT MOST. Thus, it's MORE like "not theft" than "theft" but still doesn't fit perfectly. Hammering it until it fits one pigeonhole or the other makes good soundbites, but doesn't help really answer the question. You unfortunately can't decide if it's wrong solely by how much like theft it is: you have to turn your brain on and decide for yourself if it's wrong.
Even this is not THAT big a deal. But the same problem occurs with questions like "is this murder". Some things are clearly murder. Some things are clearly NOT murder. Some things are SOMEWHAT like murder, and insisting they are or are not will tend to polarise people into "MURDER!" and "NOT MURDER!" even if they may mostly agree on everything else about it.