The two derivations are actually identical, except for the fact that Weinberg didn't have the general form of the Noether theorem for symmetries acting on the space-time coordinates as well as on the fields (Equation 2.141 in Di Francesco, Mathieu and Sénéchal's book).

As a consquence, Weinberg had to compute the variation of the action with respect to the Lorentz generators from scratch (including the substitution of the equations of motion).

Furthermor, I wanted to remark that the term depending on the variation of the space time coordinates in the general form of the Noether theorem is not due to the noninvariance of the Minkowski space time measure $d^4x$ as this measure is invariant under both translations and Lorentz transformations. The extra term is due to the dependence of the Lagrangian on the space time coordinates through its dependence on the fields.

Now, both authors use the derivation as a means of computation of the Belinfante & Rosenfeld 3-tensor whose divergence is to be added to the canonical energy momentum tensor to obtain the symmetric Belinfante energy momentum tensor. The principle upon which this computation is based is that the orbital part of the canonical conserved current corresponding the Lorentz symmetry
must have the form:

$M^{\mu\nu\rho} = x^{\nu} T_B^{\mu\rho} - x^{\rho} T_B^{\mu\nu} $

with $T_B$ both conserved and symmetric (as can be checked by a direct computation), therefore, they arrange the extra-terms they obtained to bring the Lorentz canonical current to this form and as a consequence they obtain the required tensor to be added.

I wanted to add that both authors use the derivatives of the symmetry group parameters in their intermediate computations, but this is not required. The same currents can be obtained for variation with respect to global constant parameters. If the action were locally invariant (with respect to variable parameters), then the currents would have been conserved off-shell. This is the Noether's second theorem.

Finally, I want to refer you to the this article of Gotay and Marsden describing a method of obtaining a symmetric and (gauge invariant) energy-momentum tensor directly based on Noether's theory.

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