general relativity - Locally flat coordinate and Locally inertial frame

I am having some doubts on myself regarding the above concepts in General Relativity.

First, I want to point out how I understand them so far.

A male observer follows a timelike worldline ($\gamma$) in spacetime (because he must have a proper time). He has a frame for himself.

A coordinate is a sets of numbers the observer uses to describe the spacetime in his frame (which is another way to say the spacetime in his view).

The locally flat coordinate of an observer at a time ($s\in\gamma$) is the coordinate (of his frame, of course) in which he sees the metric tensor at a neighborhood of his position be the flat metric (Christoffel symbols vanish):

$$g_{\mu\nu}(s)=\eta_{\mu\nu}$$ $$\Gamma_{\mu\nu}^\rho(s)=0$$

This coordinate depends on and is used naturally by the observer.

Now a locally inertial frame is a frame of any freely falling observer, or any observer following a geodesic ($l$). He may use or may not use the locally flat coordinate of himself. But he has a very special coordinate which is locally flat at every point is his worldline: $$\forall s\in l:$$ $$g_{\mu\nu}(s)=\eta_{\mu\nu}$$ $$\Gamma_{\mu\nu}^\rho(s)=0$$

Do I have any misunderstanding or wrong use of terminology?

Now there should be an freely falling observer $A$ (with his special coordinate) and his wordline crosses the wordline of another (not freely falling) observer $B$. And at the cross point can I believe that the two coordinate (of two frames) may be chosen to be locally identical (or equal) (that is, there exists a linear transformation locally transform one to other)?