# Why is there an extra term in definition of Noether current for spacetime translations?

I am reading Schwartz's Quantum Field Theory textbook. In chapter 3, Schwartz first defines the conserved current for a symmetry $$\phi \rightarrow \phi + \delta \phi$$ that depends on a parameter $$\alpha$$ as $$J_\mu = \frac{\partial L}{\partial (\partial_\mu \phi_n)} \frac{\delta \phi_n}{\delta \alpha}.$$ However, Schwartz later discusses the symmetry of space-time translations, and we end up with the Noether currents $$T_{\mu\nu} = \frac{\partial L}{\partial (\partial_\mu \phi_n)} \partial_\nu \phi_n - g_{\mu\nu}L.$$ This is very similar to the first definition, but with an extra term $$g_{\mu\nu}L$$. I can follow the derivation, but I'm still confused why there is this difference? It seems like the first should be a general case that would encompass the latter, but I can't exactly see how.

Another confusion I have is that the energy momentum tensor $$T_{\mu\nu}$$ has two free covariant indices, but the first term on the right hand side of the 2nd equation has one covariant index (from $$\partial_\nu$$) and one covariant index on the bottom (the $$\partial_\mu$$) which becomes a contravariant index, right? Is this an error in the book or my reasoning?

There is also a formula for when the independent variables (the $$x^\mu$$) are also varied, which has an added term, and there is also a formula when the symmetry is a quasi-symmetry, i.e. the Lagrangian is invariant only up to a total divergence.
The formula for the canonical stress energy tensor can be derived by using a vertical variation $$\delta\phi=a^\mu\partial_\mu\phi$$ ($$a$$ is a constant vector) but then this is a quasi-symmetry and the additional term comes from the total divergence, or one can use the formula for variations with nonzero horizontal components, in which case the symmetry is exact, and the additional term comes from the extended formula for the Noether current.