I came across the claim on Wikipedia here and here that there are two separate definitions or uses of the term "fictitious force", in particular the centrifugal force - one related to coordinates and the other related to non-inertial frames. More specifically, it argues that centrifugal force terms arise in polar coordinates in inertial frames.
However, I am not convinced: my argument is that there is no confusion in constructing a polar coordinate system at the origin; but there is a problem with interpretation when you introduce a polar coordinate vector basis which is a tangent vector basis along some curve which may represent the path of a particle. The polar vector basis being a tangent vector basis is then time-dependent, and this vector basis to me represents a non-inertial frame attached to the particle. Thus, when writing velocities and forces in terms of the polar coordinate vector basis we are implicitly using a non-inertial frame of reference. Frames are constructed by vector bases.
It is not clear to me if the specific claims on wikipedia are supported by related SE Q&A here, here and here.
