The concept of an external cost is like a square circle: not even wrong. Opportunity costs, the costs that actually determine economic choices, are inherently internal. Opportunity cost is the forgone opportunity of a choice. It doesn't even make sense to speak of imposing the opportunity one forgoes onto another. For example, if a firm chooses to produce another unit of smoke, forgoing the opportunity to, say, buy a snorkel, nothing about the forgone opportunity is in any sense imposed on anyone affected by the smoke. How does that even make sense? There is nothing "external" about the forgone opportunity, there is no sense in which the forgone opportunity is anything but internal.
Of course economists don't mean opportunity costs when they talk about externalities. They mean what Buchanan called pain cost, meaning that externalities happen when someone hurts someone else. Of course, that condition alone is not enough. The harms have to be external, meaning they aren't internal, meaning that the harm is in some way imposed or not chosen by the affected parties as a consequence of the best of various feasible alternatives.
It is very weird to me that this idea survived Coase's famous paper, "The Problem of Social Cost." It is hardly an obscure paper, heavily cited, practically creating the field of law and economics, and earning Coase a Nobel prize. And the lesson (or one of many) is widely taught: externalities are reciprocal and a function of the decisions of every involved party: the person who chooses to live near a smog-producing factory is as much the "cause" of the externality as the factory is. Nothing "external" is happening; people are simply choosing to bear some harm for the sake of avoiding something less desirable.
So economists know that there are no externalized harms. People choose to create externalities as the best of various alternatives, bearing some harm in the pursuit of some more-than-offsetting gain, or at least the avoidance of even worse harms. From the chooser's perspective, there is no difference between accepting the harm of smog in one's local atmosphere because she is being paid to tolerate it versus because the house is cheaper in that location, or the sunsets are very pretty, or because she likes the smell of a factory at work. The source of the compensation is irrelevant.
The idea of externalities is the idea that some of the relevant aspects of choice aren't considered by the affected individuals for no reason. This kind of sudden nonsense is endemic to welfare economics. I'll write more about that soon.