Black hole singularity from collapsing light vs dust Consider two black holes, one formed from a spherical cloud of electromagnetic radiation, and one formed from a non-interacting dust solution.
The stress energy tensor is traceless for electromagnetic radiation, and has a non-zero trace for dust.  So the Ricci curvature scalar is $R=0$ inside the collapsing cloud for electromagnetism and non-zero for dust.
Are the resulting singularities somehow distinct?  Does the geometry have an infinite $R$ at the singularity for the dust case but not the electromagnetic case, and does this "boundary" condition affect the solution outside the singularity?
 A: Off the bat, I don't know. But I do know how to find out the answer. 
In GR, one is allowed to make use of singularity theorems to classify singularities into different types. The best place to study this in my opinion is the book by Ellis and Hawking which is very mathematical but as my professor told me, "If you can work it out, then you have mastered the universe." Singularities are defined in terms of geodesic incompleteness i.e. all paths in spacetime which intersect this point, end at that point. However, GR cannot qualitatively or quantitatively describe the singularity and I think you agree with this from your comment. However, it does allow us to differentiate the kinds of singularities that may form in the universe. In fact, why singularities occur is an open problem in GR (i.e. How and why can smooth Cauchy data evolve into singular solutions?). 
We have scalar curvature singularities i.e. when at least one scalar polynomial constructed from $R_{ab}, \ g_{ab}, T_{ab}$ diverges. Such singularities occur when either the Ricci tensor or Riemann tensor or both diverge. As far as my knowledge goes, I am not sure if whether singular solutions with derivatives of these tensors are allowed to form singular solutions i.e. I don't know about the dynamic nature of singularities. In this case you can have either the Ricci scalar to be divergent or the Riemann tensor to be divergene. Divergence of $R_{ab}$ implies that the stress energy tensor is divergent and the prime example of this is the Big Bang in the FLRW cosmology.
We also have the divergence of the Riemann tensor as in the case of black hole singularities, but the Ricci tensor is regular. And as you pointed out, $R=0$ for the case of Schwarzschild singularities.
We also have the case where both $R_{ab}R^{ab}$ and  $R_{abcd}R^{abcd}$ diverge as in the case of RN black holes.
So with some of this information, all you would need to answer your problem is to check whether the Riemann tensor, Ricci tensor and stress energy tensor are regular at $r = 0$. And if not, you can differentiate them along the classification scheme I have very briefly outlined about. I should also mention that there are other possible singularities which I have not mentioned here. But if you are interested more about this, then I would recommend Ellis and Hawking's book. The complete categorisation of singularities is still an open problem.
Hope this answer is of some use to you!
A: You can have a spherically symmetric shell of dust with Minkowksi spacetime on the inside.
So as you track the surface of falling dust, it has curvature on the surface (but not a tiny bit inside the surface) and there is no curvature at the origin up until the dust surface hits it. But that's the same point when then the curvature on the dust surface goes through the roof. And the curvature already jumped at the dust surface.
If the curvature of the two is the same outside then they both blow up the way as the collapsing surface hits the singularity. But the singularity isn't a place with properties. They are places missing from the manifold that can not be put in.

the solutions are the same for r>0,

They are the same about he source and below the source. The source is collapsing, eventually the source reaches $r=0$ which is not a place.

we cannot meaningfully ask about properties of the manifold at r=0.

There is no manifold when $r=0.$

is there an actual mathematical consequence to claiming the singularity at r=0 is removed from the manifold, and so we must say it is removed?

If you tried to extend the manifold to include the excluded events, you could make manifolds that agree away from the singularity but are different manifolds. So there isn't a manifold there. And it doesn't make sense to talk about the manifold at those events.
