 To verify the categoricity of a second-order sentence one has to go through infinite structures in a way which essentially calls for set theory. The situation with internal categoricity is different. To verify the internal categoricity of a second-order sentence one just has to produce a proof. So there is a dramatic difference. And still internal categoricity is stronger than categoricity. So it would be foolish to establish categoricity if one could establish even internal categoricity. Fortunately, the classical examples of categorical sentences are all internally categorical: