The basic problem remains, though we've consolidated many problems of induction into one.
No, that's not true. The operative word is "explanation", not "short". Explanation changes everything. It is not enough to say, "The sun has risen every day in the past, therefore it will continue to rise in the future." You have to explain why the sun has risen every day in the past (i.e. the earth is round, it rotates about its axis, yada yada yada). Conciseness is not as much a requirement as it is a consequence of the more primary requirement of being a good explanation and consistent with all available data. In actual practice, these requirements generally eliminate all theories but one. It just turns out that those theories tend to be concise.
OK, on that view, yes, they are the same. But inductivism is generally understood to be more specific than that. It is generally understood to be a particular way of deriving general rules from specific observations, namely, by simply generalizing the data into patterns without any regard for causal mechanisms. So, for example, if you see (say) 100 crows and they are all black, induction tells you to conclude that all crows are black without ever asking why crows are black.