Backward differentiation formulas (BDF) are a particular kind of linear multistep methods. Such methods write [1]
\begin{aligned}
x_n &= \alpha_1 x_{n-1} + \dots + \alpha_k x_{n-k} + \beta_0 \Delta t\, f(t_n,x_n) \, .
\end{aligned}
They are designed to solve numerically ordinary differential equations
$$
x' = f(t,x) \, ,
$$
where $t_{n}-t_{n-1} = \Delta t$ denotes the step size. In particular, the BFD2 corresponds to the case $k=2$, with coefficients $\alpha_1=4/3$, $\alpha_2=-1/3$, and $\beta_0=2/3$, i.e.
$$
x_n = \frac{4 x_{n-1} - x_{n-2}
+ 2 \Delta t\, f(t_n,x_n)}{3}\, .
$$
Note that the equation for $x_n$ is implicit. BDFs are not designed to approximate derivatives, but to provide implicit schemes for ordinary differential equations. A first-order backward finite difference to approximate the second derivative
$$
d_{tt}x = \frac{x_n - 2 x_{n-1} + x_{n-2}}{{\Delta t}^2}
$$
may be more relevant in the present case.
To verify if the formula in the OP is consistent with $x''(t)$, expand $x$ in Taylor series in the vicinity of $t$, evaluate $x(t+\Delta t)$, $x(t-\Delta t)$, $x(t-2\Delta t)$ and $x(t-3\Delta t)$ until a given power of $\Delta t$, and inject these expressions in the difference formula.
[1] J.C. Butcher, Numerical Methods for Ordinary Differential equations, 2nd ed. Wiley, 2008.
