Why "frame of reference" can apparently mean two different things?

Question

There are several things about frame of reference in physics which I know.

1. We may work in $\mathbb R^3,$ but this set is annoying because it has an origin. In physics we wish to have a decentralized theory where laws are independent of the choice of origin, so we would instead work on a (Riemannian) manifold $M$ with a metric $g.$
2. A frame of reference means a one-parameter subgroup $\phi_t$ of the group of isometries of $M$. This should be different from the concept of a local coordinate frame of a manifold, which is also called a "frame" for some reason.
3. The above idea can be generalised. See this Wikipedia page.
4. However, the first sentence in this page says that a frame is a coordinate system with reference points. This definition is also very commonly seen in many books.

So, we end up with two definitions of frames of reference: (by vector I mean either tangent vector of a manifold, or simply a vector in $\mathbb R^n$)

1. The one-parameter subgroup version. A frame of reference is (at least in the definition) not a set of coordinates, but a function/an action of a Lie group that dictates how old points are changed into new ones in new frame.
2. The coordinate version. A frame of reference is a system of coordinates that describes the system. Saying that it is a change of coordinates in some way implies that (tangent) vectors/points themselves should be "independent" of the coordinate/frame in some sense (not always "independent" - the acceleration vector in rotating frame is different from that in an inertial frame). This is very different from the first definition, where vectors themselves are changed.

So, why do we have two very different meanings for "frame of reference"? In practical application of frames, we never seem to run into this problem.