I would argue that the difference in how the deductive-inductive dichotomy is framed in terms of premises and conclusions and how Pierce's characterization function is a result of the differences in how both topics relate to logic and epistemology and the two senses of argument. In the narrow sense, an argument is an inference from two or three sentences, where as in the broad sense, an argument is a structure of narrow arguments as well as other objects that attempts to achieve some end. To prove Socrates is mortal because he is a man is an example of the first, to argue that human consciousness resembles a Turing machine would exemplify the second. This is mirrored in the fact that logic as the ancient Greeks saw it was more epistemological and less ontic in nature.
Deductive reason which started with very simple systems like Aristotelian syllogism has always been of limited scope with the focus being on the certainty of statements in regard to each other; here the focus is on almost grammatical-level examination of what constitutes truth about simple entities and their properties. How do Socrates, man, and mortality relate? This would the analytic distinction you mention.
Yet, in informal argumentation, where the emphasis is generally not argumentation for the sake of understanding inference and argumentation, but rather interested in the messy, informal, and complicated business of reality, the emphasis is on producing an argument through inference that works. This would be the non-analytic distinction you cite.
From the perspective of someone who is more interested in the epistemic aspect of argumentation rather than the logical aspect, it would be natural to place an emphasis not on the level of statement and proposition which are one level of meaning-bearer, but rather on a higher, more holistic level: cases and the rules that can be derived from them. Ray Jackendoff proposes in his Foundations of Language that meaning occurs at multiple levels: the phonological, syntactic, semantic, and spatial levels, so that the argument to prove Socrates mortality is essentially grammatical, whereas the argument concerned with the adequate characterization of mind would be at the highest level possible. A concrete example may serve to illustrate.
In analogical argumentation such as that used in legal reasoning by case, one is not about the nuts and bolts of the grammar generally. Certainly, disputes may over occur in regards to the meaning of statements by parties, but the overall concern is how to rule in regards to the conflict, let us say in civil law. A trier of fact must resolve by deciding what to do, and in the Anglo-American tradition of law, this creates a precedent or rule for future judges. So, the distinction in paradigms reflects the notion that there are two subtly different definitions of argument, one in the narrow technical sense, and one in the broader practical sense.