Types of singularities include curvature singularities and conical singularities. So, for a curvature singularity(black hole) with geodesic incompleteness, is it the same as a physical singularity? If there is something wrong with the following contents, please correct it.
I think you have the types of singularities slightly mixed up (although possibly I have them mixed up, so watch out!)
All singularities, I think. involve geodesic incompleteness: there's a notion that you can't extend a geodesic to arbitrarily large values of its affine parameter because something goes wrong. So, in particular the singularity theorems say things like 'given some preconditons on causality, energy and gravity, there will be geodesic incompleteness'.
Two subtypes of singularities are curvature singularities, where physical properties blow up as you approach the singularity (and 'become infinite' at the singularity itself) and conical singularities, where things don't blow up as you approach the singularity. (There's another subtype which is the 'I just decided to chop some points out of spacetime' type, but we can ignore that.)
So then I think that it is expected that the singularities at the centre of black holes will be curvature singularities, while cosmic strings, if they exist, would be conical singularities. But the singularity theorems don't actually talk about what sort the singularities they predict are: they just say there will be geodesic incompleteness.