Actually, the metric variational definition for the stress-energy tensor is an universal *improvement* procedure for the canonical stress-energy tensor (and hence not always concides with the latter), in a sense which will be made precise below. Such a procedure is necessary because the canonical stress-energy tensor, although always conserved, often fails to satisfy other physical requirements like gauge invariance (since it is an observable quantity), symmetry (needed if we want it to be a source for the gravitational field) and tracelessness (for locally scale invariant theories). For example, all three requirements fail for pure electrodynamics in four space-time dimensions.

Even if you are dealing with a field theory in Minkowski space-time, it is inevitably coupled to gravity simply because of the fact that the Lagrangian depends on the space-time metric (here taking the particular value of the Minkowski metric). The particular dynamics of the metric is irrelevant - all we need is that there are no other "external" fields besides the metric and that the field action functional is diffeomorphism invariant.

Let $L=L(\phi,g)$ be a local field Lagrangian in the space-time $(M,g)$, and $$S_K[\phi,g]=\int_K L(\phi,g)\sqrt{|\det g|}\mathrm{d}x\ ,\quad K\subset M\text{ any bounded region}$$ the corresponding (family of) action functional(s indexed by $K$ as above). We allow $L$ to have finite but otherwise arbitrary order dependence on $\phi$ and $g$, and no explicit space-time dependence since we want it not to depend on any other fields. The infinitesimal variation of $S_K$ with respect to a vector field $X$ on $M$ (i.e. an infinitesimal diffeomorphism) is then given by $$\delta_X S_K[\phi,g]=\int_K\left(\frac{\delta L(\phi,g)}{\delta g_{\mu\nu}}\delta_X g_{\mu\nu}+\frac{\delta L(\phi,g)}{\delta \phi^j}\delta_X \phi^j+\nabla_\mu(T^{\mu\nu}X_\nu)\right)\sqrt{|\det g|}\mathrm{d}x\ ,\quad X_\rho=g_{\rho\sigma}X^\sigma\ ,$$ where $\frac{\delta L(\phi,g)}{\delta g_{\mu\nu}}$ and $\frac{\delta L(\phi,g)}{\delta \phi^j}$ are respectively the Euler-Lagrange (i.e. variational) derivatives of $L(\phi,g)$ with respect to $g$ and $\phi$, $\nabla$ is the Levi-Civita covariant derivative associated to $g$, $T^{\mu\nu}$ is the (canonical or improved) stress-energy tensor, $$\delta_X g_{\mu\nu}=\nabla_\mu X_\nu+\nabla_\nu X_\mu$$ is the Lie derivative of $g$ along $X$ and $\delta_X\phi^j$ depends on the particular way we lift $X$ to a projectable vector field on the total space of the fiber bundle over $M$ where the fields $\phi^j$ live (for instance, if they are all scalar fields, we simply have $\delta_X\phi^j=-X\phi^j=-X^\mu\nabla_\mu\phi^j$). 

There is an implicit but crucial requirement on the admissible improvements for $T^{\mu\nu}$ - namely, the improved Noether current $j^\mu(L,X)=T^{\mu\nu}X_\nu$ should not only be linear in $X$ but depend *only on the point values of* $X$ (we call this property *ultralocality*). This requirement also affects to a certain extent the definition of the infinitesimal field variation $\delta_X\phi^j$, but the details of this are not important in what follows. Why do we insist on this requirement? As we shall see below, ultralocality singles out a unique improvement prescription for $T^{\mu\nu}$ which in addition satisfies all physical desiderata.

Diffeomorphism invariance of the action functional means we require that $\delta_X S_K[\phi,g]=0$ *for all* $X,\phi,g,K$. If, in addition, the fields $\phi^j$ satisfy the Euler-Lagrange equations of motion, we have that $$2\frac{\delta L(\phi,g)}{\delta g_{\mu\nu}}\nabla_\mu X_\nu+\nabla_\mu(T^{\mu\nu}X_\nu)=\left(2\frac{\delta L(\phi,g)}{\delta g_{\mu\nu}}+T^{\mu\nu}\right)\nabla_\mu X_\nu+X_\nu\nabla_\mu T^{\mu\nu}=0\ .$$ Since $X$ is arbitrary and therefore we may specify $X_\nu$ and $\nabla_\mu X_\nu$ independently at each point of $M$, we obtain in a single stroke:

 1. The desired variational formula for the *improved* stress-energy tensor $$T^{\mu\nu}=-2\frac{\delta L(\phi,g)}{\delta g_{\mu\nu}}$$ and therefore the symmetry $T^{\mu\nu}=T^{\nu\mu}$;

 2. The covariant conservation law $\nabla_\mu T^{\mu\nu}=0$;

 3. If the metric happens to obey a dynamics determined by a Lagrangian $L_G(g)$, then $T^{\mu\nu}$ automatically becomes the source to the metric equations of motion. This also guarantees compliance with the second Noether theorem, as it should - the canonical Noether current associated to the *total* (i.e. metric + field) Lagrangian and to $X$ still vanishes on shell if the total action functional is also diffeomorphism invariant.

Although it is not trivial to show, $T^{\mu\nu}$ also happens to be traceless if the field theory exhibits local scale invariance.

If the fields $\phi^j$ are all scalar and $L(\phi,g)$ does not depend on derivatives of $g$, then $T^{\mu\nu}$ coincides with the canonical stress-energy tensor. This is *no longer* the case for spinor fields, whose Lagrangian usually also depends on the first derivatives of the metric through the spin connection, for scalar fields with non-minimal curvature coupling, or for the electromagnetic field.

The above understanding of the metric variational definition of the stress-energy tensor in full generality came surprisingly late - it was thoroughly developed by M. Forger and H. Römer ("Currents and the Energy-Momentum Tensor in Classical Field Theory: a Fresh Look at an Old Problem". Ann.Phys. **309** (2004) 306-389, [arXiv:hep-th/0307199][1]), whose work we warmly recommend for (many) more details and examples.

  [1]: http://arxiv.org/abs/hep-th/0307199