The words "in general" in general refer to the general case. In general is not "in specific", it's "in general". In general, in the Lagrangian formulation (as cited repeatedly in the article) one finds the centrifugal force using the definition in terms of , with no restrictions upon the "highly abstract Lagrangian coordinates, far removed from anything". That is not beside the point, that is the literature of several subject areas. 

You are confusing two disparate conceptions here: "the fictitious/ inertial forces arising in terms of a rotating frame"  are absolutely not  "just a special case of the more general fictitious/forces that arise whenever a system of reference is used in which the spatial coordinates of inertial paths are not linear functions of the time coordinate." The former are used to define inertial frames and are zero in such frames; the latter are defined regardless of the frame, vary with the coordinate system regardless of the frame, and may well be non-zero in inertial frames.