Doubt in the definition of the stress-energy tensor in Peskin and Schroeder's QFT book

We can describe the infinitesimal translation
$$
x^{mu} to x^{mu}-a^{mu}
$$
alternatively as a transformation of the field configuration
$$
phi(x) to phi(x+a) = phi(x) + a^{mu} phi(x).
$$
The Lagrangian is also a scalar, so
$$
mathcal{L} to mathcal{L} + a^{mu} partial_{mu}mathcal{L} = mathcal{L} a^{nu} partial_{mu} (delta_{nu}^{mu} mathcal{L}).
$$

If we compare this to equation
$$
mathcal{L}(x) to mathcal{L}(x) + alpha partial_{mu} mathcal{J}(x)
$$
we see that we now have a nonzero $mathcal{J}^{mu}$.

Up to this point, I basically copied Peskin and Schroeder’s book "An Introduction to QFT". Then, they state: "Taking this into account, we can apply the theorem to obtain four separately conserved currents:"

$$
T^{mu}_{nu} equiv frac{partial mathcal{L}}{partial(partial_{mu} phi)} partial_{nu} phi – mathcal{L} delta^{mu}_{nu}
$$
this last equation corresponds to (2.17) in the book, but I really cannot see how we get there. The fact that $mathcal{J}$ is a vector (rank-1 tensor), not a tensor (rank-2 tensor), as the SET was defined, stands as my main issue.