Dear Jenkins, the theories you want to construct are "noncritical string theories" and they're less interesting and less consistent than the "critical string theories".
First, the Nambu-Goto action - the proper area of the world sheet - is nonlinear. It includes square roots etc. It's much better to introduce an auxiliary metric tensor on the world sheet and the action for the coordinates $X$ becomes nice and bilinear - a free theory.
However, we don't want new degrees of freedom to be added. The 2D metric tensor has three independent components. Two of them may be set to a standard form by the 2 degrees of freedom in the 2D coordinate reparameterization symmetry; and the third by the Weyl symmetry if it exists.
If it doesn't exist, it's too bad. The auxiliary world sheet metric may only be brought to the form of $e^\phi \eta_{ab}$. That means that $\phi$, determining the overall scaling, becomes another function of the world sheet coordinates $(\sigma,\tau)$, very analogously to the spacetime coordinates $X(\sigma,\tau)$. In fact, it is really valid to say that the parameter determining the overall scaling of the metric is another spacetime coordinate.
If this coordinate were totally identical to the other coordinates, then there would also be a translation symmetry in the $\phi$ direction - but that's equivalent to the Weyl symmetry (multiplicative scaling of $e^\phi$ is the same thing as additive shifts to $\phi$). Because by assumption, the Weyl symmetry doesn't hold in your theory, the new spacetime coordinate $\phi$ can't have quite the same properties as the other spacetime coordinates.
However, in normal circumstances, you obtain the violations of the Weyl invariance as a disease. In particular, if you try to study string theory in a non-critical dimension, i.e. $D\neq 26$ or $D\neq 10$, you will find out that the field $\phi$ doesn't decouple and the path integral, when calculated including the one-loop accuracy, still depends on $\phi$. So the Weyl symmetry, equivalent to an additive shift of $\phi$ by a function of the world sheet, is not a symmetry.
As I said, this can be interpreted as $\phi$'s becoming a new spacetime coordinate. But if you try to calculate the effective action in the new spacetime that has an additional dimension $\phi$, you will find out that the laws of physics are not invariant under translations in $\phi$ - that's nothing else than the failure of the theory to be Weyl-invariant.
In particular, you will find out that the dilaton linearly depends on $\phi$: search for papers about "linear dilaton". The squared gradient of the dilaton is related to the surplus or excess (if it is time-like or space-like) of the spacetime coordinates, relatively to the critical dimension.
If the spacetime has two dimensions, one may choose the dilaton to depend on the (only) spacelike coordinate $\phi=X^1$ in such a way that the theory including $\phi$ is Weyl-invariant again. In this case, it's useful to consider not only the right linear dilaton - solving the equations of motion - but also a non-trivial background for the tachyon. One ends up with the so-called "Liouville theory" - a "linear dilaton" theory with some extra tachyonic profile in a non-critical stringy $D=2$ spacetime - which is slightly more consistent than other noncritical string theories. The Liouville theory may also be described by a quantum mechanical model with a large matrix - the old matrix theory.
OK I think I know what is going on. It's all about primes. Consider an active spacetime transformation:
$$ x^{\mu} \mapsto x'^{\mu}(x) \, ,$$
$$g_{\mu \nu} (x) \mapsto g'_{\mu \nu} (x') = g_{\alpha \beta} (x) \frac{\partial x^{\alpha}}{\partial x'^{\mu}} \frac{\partial x^{\beta}}{\partial x'^{\nu}} \, .$$
(the transformation of the metric tensor follows from the fact that it is a rank 2 tensor). With this notation both Di Francesco and David Tong are wrong (as far as I understand). The GR book by Zee on the other hand writes it properly. First of all consider an isometry. This is an spacetime transformation as before that leaves the metric invariant, meaning
$$ g'_{\mu \nu} (x') = g_{\alpha \beta} (x) \frac{\partial x^{\alpha}}{\partial x'^{\mu}} \frac{\partial x^{\beta}}{\partial x'^{\nu}} = g_{\mu \nu} (x') \, .$$
(watch the primes). On the other hand a conformal transformation is a transformation that satisfies a weaker condition: it leaves the metric invariant up to scale, meaning
$$ g'_{\mu \nu} (x') = g_{\alpha \beta} (x) \frac{\partial x^{\alpha}}{\partial x'^{\mu}} \frac{\partial x^{\beta}}{\partial x'^{\nu}} = \Omega^2(x')g_{\mu \nu} (x') \, .$$
Now there should be no inconsistency. Di Francesco's definition was wrong (according to this convention/notation/understanding) because it compared the metric before and after the transformation at different points, and you have to compare them at the same point.
Best Answer
The Weyl transformation and the conformal transformation are completely different things (although they are often discussed in similar contexts).
A Weyl transformation isn't a coordinate transformation on the space or spacetime at all. It is a physical change of the metric, $g_{\mu\nu}(x)\to g_{\mu\nu}(x)\cdot \Omega(x)$. It is a transformation that changes the proper distances at each point by a factor and the factor may depend on the place – but not on the direction of the line whose proper distance we measure (because $\Omega$ is a scalar).
Note that a Weyl transformation isn't a symmetry of the usual laws we know, like atomic physics or the Standard Model, because particles are associated with preferred length scale so physics is not scale invariant.
On the other hand, conformal transformations are a subset of coordinate transformations. They include isometries – the genuine geometric "symmetries" – as a subset. Isometries are those coordinate transformations $x\to x'$ that have the property that the metric tensor expressed as functions of $x'$ is the same as the metric tensor expressed as functions of $x$. Conformal transformations are almost the same thing: but one only requires that these two tensors are equal functions up to a Weyl rescaling.
For example, if you have a metric on the complex plane, $ds^2=dz^* dz$, then any holomorphic function, such as $z\to 1/z$, is conformally invariant because the angles are preserved. If you pick two infinitesimal arrows $dz_1$ and $dz_2$ starting from the same point $z$ and if you transform all the endpoints of the arrows to another place via the transformation $z\to 1/z$, then the angle between the final arrows will be the same. Consequently, the metric in terms of $z'=1/z$ will be still given by $$ ds^2 = dz^* dz = d(1/z^{\prime *}) d (1/z') = \frac{1}{(z^{\prime *}z')^2} dz^{\prime *} dz' $$ which is the same metric up to the Weyl scaling by the fraction at the beginning. That's why this holomorphic transformation is conformal, angle-preserving. But a conformal transformation is a coordinate transformation, a diffeomorphism. The Weyl transformation is something else. It keeps the coordinates fixed but directly changes the values of some fields, especially the metric tensor, at each point by a scalar multiplicative factor.