special relativity - Why doesn't time change in the non-relativistic limit of Lorentz transformations?

A simple boost in the $x$ direction is given by:

$$\Lambda = \begin{pmatrix} \cosh(\rho) & \sinh(\rho) & 0 & 0 \\ \sinh(\rho) & \cosh(\rho) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}$$

Which get linearized to the following transformation:

$$x^0 \mapsto x^0 , \quad x^1 \mapsto x^1 + \frac vc x^0$$

How come the zeroth component is not linearized to $x^0 \mapsto x^0 + \frac vc x^1$? Is that because there is another factor $c$ in the time components? Since $x^0 = ct$, that would mean the time is transformed like

$$t \mapsto t + \frac v{c^2} x,$$ and $c^{-2}$ is just so small that is ignored?

Or is it just to fit the Galilei transformation?