Energy-momentum tensor for a Lagrangian with no explicit spatial dependence

Suppose I have a Lagrangian $mathcal{L} = mathcal{L}(phi, partial_mu phi)$ for a (let’s say) real scalar field with no explicit temporal or spatial dependence. Then I believe the usual Noether computation gets us that (for an infinitesimal symmetry with $delta mathcal{L} = 0$) that

$$X^mu = frac{partial mathcal{L}}{partial (partial_mu phi)} delta phi $$

satisfies a continuity equation $partial_mu X^mu = 0$. For the case of spatial translation $(x’)^mu = x^mu + a^mu$ with $phi’ = phi + (partial_nu phi) a^nu$ we get

$$X^mu = frac{partial mathcal{L}}{partial(partial_mu phi)} (partial_nu phi) a^nu$$

and since this is valid for any vector $a^nu$ then (I believe) we can conclude the energy-momentum tensor

$$T^mu_nu = frac{partial mathcal{L}}{partial(partial_mu phi)} partial_nu phi$$

satisfies $partial_mu T^mu_nu = 0$ for every $nu$. I would like to know if this account is correct, especially because when I try to apply it to the Lagrangian for a one-dimensional wave

$$ mathcal{L} = frac{(partial_0 phi)^2}{2} – frac{(partial_1 phi)^2}{2}. $$

I get as momentum $T^0_1 = (partial_0 phi)(partial_1 phi)$ which is kind of unexpected for me (I was expecting to have just $partial_1 phi$).