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I like your definitions, but all three of these could be called subsets of compilers.

By the definitions given they can not, as no function subsumes another. By whatever you define as "compiler" maybe, but I see no point in this kind of interaction that essentially boils down to subsumtion to an entity you refuse to describe any further.

Is there a merit to this? Can whatever you call compiler do more? Is it all three of the things mentioned combined? Who knows - as is stands I only know that you disagree with the definitions given/proposed.

I think they are fine definitions. I think a transpiler, a term rewriter, an assembler, a stand-alone optimizer, and even some pretty printers are subclasses of compilers.

I define a compiler as something that takes an input in a language, does transformations, and produces a transformed output in a language. All of them do that, and they are more specific terms for types of compilers.

Except that they do what useful words do; provide (more) useful information.
Fair. I don’t believe I said they were useless terms for differentiation of types of compilers, though. I just said they can all be thought of as a class as different types of compilers.
So what? A car, a bike, and a truck can all be called subsets of vehicles, but we still have (and need) different words for each type.
Primates, canids, felids, and ungulates are all subsets of mammals and all have further subsets. Mammalia is further a subset of animalia. When we’re discussing categorizations, it’s often helpful to have multiple levels of categories. I’m not sure why you seem to be calling out specificity as a gotcha, when my argument isn’t at all that we don’t need multiple terms. It’s that we should consider these things in terms of similarity and specific differences, not throw away a term as useless as the article and its headline suggest.

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