Thanks for this. Part of what I like about researching histories is finding shadowy old paths covered in briars that weren't originally explored (in addition to re-treading the ancestral path and plotting its trajectory on a map).
When I attempt an explication a la Carnap (which reminds me much of Frege), it seems I have a number of possible routes. The most straight forward is of a somewhat discrete unpacking of a box of christmas ornaments. Once they're all laid out, I can decide which to decorate the tree with. This is useful for math proofs, where terms are often just shorthand or compositions of shorthands. For example, saying that a function is continuous can be expanded to a statement of the Bolzano-Weierstrass definition when in the context of real analysis. The statement "a holomorphic function is analytic" is an example of a 'composition,' or nested shorthand, which can be unpacked piece by piece.
Another route is the one I find amenable to analysis of ordinary language, and is a continuous unpacking, more like a 'zooming in.' The explicandum is first as if it were far away and visible as a meaningful, if somewhat mysterious, whole. Embarking on the explication brings one closer to it gradually, revealing increasing levels of detail until I'm satisfied ("closer" is somehow not quite right; it's a falling in, a reaching out, an assimilation). It may conclude swiftly, almost automatically, or I may grasp around blindly and in vain at ever expanding swirls of images and sounds. Complicated, emotional concepts tend to take this route. In a way, the concept is already "exact," but it's exactness is not perceived.
The last approach that comes to mind, and the most difficult, is a modular-constructive approach, wherein tenuously related concepts may be explored in hopes of finding similarities. Small pieces that might be useful are glued together and wired up in various ways, and one tests to see how the resulting device works at each stage, sizing up new additions and assessing how it's progressing towards the goal. This is useful when one seeks a causal narrative of an observed phenemona.