Does the light-cone at every point plus a scale uniquely define a spacetime metric? In general relativity, knowing the local light-cone shape at each point (i.e. the tilt, and angles of the light cone) doesn't seem enough to get the metric. Do we also need a scale as well at each point? Because multiplying the metric by a scale factor will give the same light cone at each point in space and GR is not scale invariant.
But if we knew both the shape of the lightcone at each point in spacetime plus a scale factor at each point in space-time, is this enough to know the complete pseudo-Riemannian geometry of spacetime?
Secondly if visualising this in a diagram, how would we depict this scale factor?
 A: As mentioned in the comments, two metrics related by a conformal transformation have the same light-cone.
What is the scale factor equivalent to? It's time to bring up the topic few people discuss, measurement in GR.
There are a variety of structures on a spacetime manifold. A structure helps us define an equivalence relation between two spacetimes. If we have two spacetimes $M$, $M'$, and a diffeomorphism $\phi : M \to M'$, we say that those two spacetimes have the same structure if it preserves some property : 


*

*Two spacetimes are isometric if $\phi_* g = g'$. This is the metric structure.

*Two spacetimes are causally isomorphic if, for $p < q$ (resp. $p \ll q$), $\phi(p) \ll \phi(q)$ (resp. $\phi \ll \phi(q)$)0. This is the causal structure.

*Two spacetimes are conformally equivalent if they are related by a Weyl transform, so that there exists a function $\omega$ for which $\phi_* g = e^{\omega} g'$.

*Two spacetimes are affinely equivalent if, for any geodesic $\gamma$, we have that $\gamma' = \phi \circ \gamma$ is a pre-geodesic (ie $\nabla'_{\dot{\gamma}'} \dot{\gamma}' = f \dot{\gamma}'$).


Mathematically, the structure is an equivalence class, so that a structure on the metric is the equivalence class $[g]$, etc. Some of those classes are related somewhat : If a spacetime is distinguishing, the causal structure is equivalent to the conformal structure, for instance.
Different measurements give us informations about different structures. A common measurement scheme is the EPS approach, which roughly measures quantities based on the order of events (ie whether some experiment will give us a response before or after another). In particular, it does not involve informations about clocks, or proper times. Due to its construction, the EPS approach only gives us the affine and conformal structure of a spacetime, and not its metric. The measurements will give us a metric, but not the metric, simply one related to the "real" one via one of those equivalences. Adding proper time to the informations of the EPS approach will give you the actual metric (up to diffeomorphism).
So let's see what's the influence of a conformal transformation on proper time : 
\begin{eqnarray}
\tau &=& \int_{t_1}^{t_2} \sqrt{e^{\omega(\gamma(t))} g_{\gamma(t)}(u(t), u(t))} dt\\
&=& \int_{t_1}^{t_2} e^{\frac{\omega(\gamma(t))}{2}} \sqrt{g_{\gamma(t)}(u(t), u(t))} dt
\end{eqnarray}
If we switch to an affine parametrization in the original spacetime, this is simply 
\begin{eqnarray}
\tau &=& \int_{t_1}^{t_2} e^{\frac{\omega(\gamma(t))}{2}} dt
\end{eqnarray}
Now let's consider this : around a point $p$, take a sequence of curves with affine parameters, $\gamma(0) = p$, by restricting their domains to $[-\varepsilon, \varepsilon]$. We have : 
\begin{eqnarray}
\tau_\varepsilon &=& \int_{-\varepsilon}^{\varepsilon} e^{\frac{\omega(\gamma(t))}{2}} dt
\end{eqnarray}
If we pick some function $F(t)$ such that $F'(t) = \exp(\omega(\gamma(t)) / 2)$, this is 
\begin{eqnarray}
\tau_\varepsilon &=& F(\varepsilon) - F(-\varepsilon)
\end{eqnarray}
Therefore, we have that $\tau_\varepsilon / 2\varepsilon$ converges to $\exp(\omega(p) / 2) $ : the conformal factor rescales the proper length of curves. Locally, a proper time will be longer by a factor of the square root of the conformal factor around that point (remember that as we picked affine parametrization for our original curve, $2\varepsilon$ is indeed the original proper time). You can see this a bit better by considering the Taylor expansion around $0$ : 
\begin{eqnarray}
e^{\frac{\omega(\gamma(t))}{2}} = e^{\frac{\omega(p)}{2}} [1 + t d\omega_p[\dot{\gamma}(0)]  + \mathcal{O}(t^2) ]
\end{eqnarray}
If we use Fermi coordinates of our curve, so that $\partial_t = \dot{\gamma}$, 
\begin{eqnarray}
e^{\frac{\omega(\gamma(t))}{2}} = e^{\frac{\omega(p)}{2}} [1 + t \partial_t\omega(t, \vec{x})|_{(t, \vec{x}) = 0}  + \mathcal{O}(t^2) ]
\end{eqnarray}
Therefore, our proper time is 
\begin{eqnarray}
\tau &=& \int_{t_1}^{t_2} e^{\frac{\omega(p)}{2}} [1 + t \partial_t\omega(t, \vec{x})|_{(t, \vec{x}) = 0}  + \mathcal{O}(t^2)] dt\\
&=& e^{\frac{\omega(p)}{2}} [ (t_2 - t_1) + \frac{(t_2 - t_1)^2}{2} \partial_t\omega(t, \vec{x})|_{(t, \vec{x}) = 0}  + \mathcal{O}((t_2 - t_1)^3)]
\end{eqnarray}
This also works for spacelike curves, but for obvious reasons, all null curves are conformally invariant. This applies to a variety of measurements. Volumes in particular are also scaled up. If you wish to represent the conformal transformation, this is the way to do it. The derivatives of the transformation will imply different scalings in different directions, as well. 
A: 
Proposition. Given a pseudo-Riemannian manifold $M$ of mixed$^1$ signature with 2 metric structures that yield the same lightcone/causal structure. Then the 2 metric tensors are related by a scale factor.

Pedestrian proof using local coordinates: Consider an arbitrary point $p\in M$. Use Riemann normal coordinates (which exist locally on a pseudo-Riemannian manifold) to bring the first metric at $p$ on the form
$$\eta_{\mu\nu}~=~{\rm diag}(\underbrace{1,\ldots,1}_{r\geq 1},\underbrace{-1,\ldots,-1}_{s\geq 1}).\tag{1}$$
Call the second metric
$$g_{\mu\nu}~=~\begin{pmatrix} g_{ij} & g_{ib}\cr g_{aj} & g_{ab}\end{pmatrix},\qquad i,j~\in\{1,\ldots,r\},\qquad a,b~\in\{1,\ldots,s\}.\tag{2}$$
We now diagonalize the real symmetric matrices $g_{ij}$ and $g_{ab}$ with $O(r)$ and $O(s)$ coordinate transformations, respectively, without spoiling eq. (1).
Consider next a null/lightlike vector
$$X^{\mu}~=~\begin{pmatrix} e^i\cr f^a\end{pmatrix},\qquad i~\in\{1,\ldots,r\},\qquad a~\in\{1,\ldots,s\},\tag{3}$$
where
$$\begin{align}0~=~&X^{\mu}\eta_{\mu\nu}X^{\nu}
~=~\sum_{i=1}^r e^ie^i-\sum_{a=1}^s f^af^a\cr
\qquad&\Rightarrow\qquad \sum_{i=1}^r e^ie^i~=~\sum_{a=1}^sf^af^a, \end{align}\tag{4}$$
and
$$0~=~X^{\mu}g_{\mu\nu}X^{\nu}~=~\sum_{i=1}^r e^ig_{ii}e^i +2\sum_{i,a} e^ig_{ia}f^a + \sum_{a=1}^sf^ag_{aa}f^a. \tag{5}$$
Eq. (4) also holds for $e^i\to -e^i$, so eq. (5) separates into
$$ \sum_{i,a} e^ig_{ia}f^a~=~0\qquad\Rightarrow\qquad g_{ia}~=~0,\tag{6} $$
and
$$\sum_{i=1}^r e^ig_{ii}e^i~=~-\sum_{a=1}^sf^ag_{aa}f^a.\tag{7}$$
Finally compare eqs. (4) and (7). By trying out different null/lightlike vectors (3), we conclude that
$g_{ij}$ and $g_{ab}$ are both proportional to a unit matrix (and with opposite proportionality factors).
Hence the 2 metrics $\eta_{\mu\nu}$ and $g_{\mu\nu}$ are proportional at the point $p$. Since the 2 metric structures are smooth structures the proportionality factor must be a smooth function. $\Box$
References:

*

*S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time, p. 61.

--
$^1$ We obviously have to exclude the Riemannian case as it has no lightcone.
A: Not quite. You need also to define a time axis at each point, because the light cone is the same for all local inertial frames at that point. Once you have that, you can determined spacetime structure from the scale on the time axis.
One way you might do this diagramatically, at least within a region of spacetime, is to restrict to two dimensions and define coordinates such that the speed of light is constant, i.e. light is shown by straight lines (typically one choses 45 degrees). Then one has a natural direction for the time axis (vertical) at each point, and one can describe the spacetime by the time scaling factor at each point.
