Transmission coefficient and transmission probability

I think at least in the case of tunneling of particles through potential barriers, the transmission coefficient and the transmission probability are the same. This make sense since $$R + T = 1$$ where $R$ and $T$ are the reflection and the transmission coefficients respectively. Also the following are always true $$R, Tge 0,~text{and}~ R, T le 1$$ The two above formula are postulates of probability measures in mathematics.

In the context of scattering theory, Transmission coefficient and transmission probability usually mean the same thing, but not the same thing as the transmission amplitude.

Transmission coefficient is however are "more correct" term, as it is defined via the scattering matrix. Interpreting it as a probability may be tricky, as it implies that we need to define the initial and the final states between which the transition occurs or something similar.

Note also that these terms are specific to one-dimensional applications of the scattering matrix formalism - in higher dimensions we cannot meaningfully distinguish transmission and reflection, but rather speak about scattering at certain angle (again, it is not uncommon to say probability fo scattering at a certain angle).