The fundamental quantity in thermodynamics is entropy, which is a function of $n$-variables $S=S(x_1, x_2,...,x_n)$. For instance, for a simple mono-component system $S=S(U,V,N)$ where $U$ is internal energy, $V$ is volume, and $N$ composition.
Taking the differential
$$\mathrm{d}S = \sum_i \left( \frac{\partial S}{\partial x_i}\right)_{j \neq i} \mathrm{d}x_i = \sum_i F_i \mathrm{d}x_i$$
The quantities $F_i \equiv ({\partial S}/{\partial x_i})_{j \neq i} $ are intensive entropic parameters and measure the change in entropy when variables change. For instance the intensive entropic parameter $(1/T)$ gives the change on entropy due to a change in the energy $U$.
Using the thermodynamic theory of fluctuations it can be shown that
$$F_i = k \left( \frac{\partial \ln P}{\partial x_i}\right)$$
where $P$ is the probability of a fluctuation in the variables near an equilibrium state.
Using the definition of average
$$\langle A \rangle = \int A P \mathrm{d}x_1 \mathrm{d}x_2 \cdots \mathrm{d}x_n$$
the demonstration of the central result of linear nonequilibrium thermodynamics
$$\langle F_i \cdot x_j \rangle = -k \delta_{ij} $$
is direct although it needs first the use of $(\partial \ln P / \partial x_i)P = \partial P / \partial x_i$ in the integrand and next integration by parts.
What Wikipedia makes is to rewrite this central result using the new quantities $X_i \equiv - F_i / k$ but the physically important quantities are the $F_i$ often also called in this context thermodynamic forces or affinities.