What is the most general expression of Noether's current in classical field theory?

Peskin & Schroeder’s expression of the Noether current If a (quasi-)symmetry is defined as a transformation that changes the action by a surface term i.e. $$Sto S’=S+int d^4x partial_mu K^mu(phi_a),tag{1}$$ or equivalently, the Lagrangian changes by a 4-divergence, $$mathscr{L}tomathscr{L}’=mathscr{L}+partial_mu K^mu,tag{2}$$ then considering transformations on the fields only, the expression of the Noether current turns out to be $$j^mu(x)=sumlimits_afrac{partial mathscr{L}}{partial(partial_muphi_a)}deltaphi_a -K^mu.tag{3}$$ P & S give the example of an internal transformation where $K^mu=0$, and a spacetime transformation (namely, spacetime translation) under which $K^muneq 0$.

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Lewis Ryder’s expression of the Noether current Here, the symmetry of the action is defined as a transformation that leaves the action invariant i.e. $$Sto S’=S.tag{4}$$ Considering transformations $$x^muto x^{mu’}=x^mu+delta x^mu, phi(x)tophi'(x)=phi(x)+deltaphi(x),$$ they derive the following expression for the Noether current $$j^mu(x)=sumlimits_afrac{partial mathscr{L}}{partial(partial_muphi_a)}deltaphi_a -Theta^{munu}delta x_nutag{5}$$ where $Theta^{munu}$ is the stress-energy tensor given by $$Theta^{munu}=sumlimits_afrac{partial mathscr{L}}{partial(partial_muphi_a)}partial^nuphi_a -eta^{munu}mathscr{L}.tag{6}$$

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Question $1$ Between $(3)$ and $(5)$ which expression of the Noether’s current is more general?

Question $2$ By generalizing Ryder’s definition of symmetry $(4)$ (to a quasi-symmetry, i.e., $(1)$), we will obtain $$j^mu(x)=sumlimits_afrac{partial mathscr{L}}{partial(partial_muphi_a)}deltaphi_a -Theta^{munu}delta x_nu-K^mu.tag{7}$$ Should $(7)$ be regarded as the most general expression of the Noether current?