Why is everyone so interested in nitpicking semantics

This is not nitpicking semantics. This is looking at the actual things that are not merely being named by the words, but described at length in the cited sources: selecting only the evidence one likes, versus accepting the evidence whether one likes it or not. When Eliezer talks about rationalisation, he is not talking about hypothetico-deductivism. Gelman and Shalizi talk about hypothetico-deductivism; they do not talk about rationalisation. Their respective definitions do not at all "match up closely".

In that particular instance, the inexactness of his language means it is possible Eliezer meant to use the term under the narrow confines of his example,

There is nothing inexact about his language, and his example is not a narrow confine.

but several commenters on that post seemed to be using my definition

Your definition bears no relationship to anyone else's use of the word "rationalisation". No definition of the word includes the idea of testing a hypothesis; all definitions relevant to the present context include the idea of fitting the evidence to the hypothesis.

there can be no doubt that Eliezer favors inductive inference, and that the paper favors hypothetico-deductivism.

Indeed so, and he has had some things to say about hypothetico-deductivism. But not in the cited article, and he does not call it rationalisation.