I think most people would agree with this definition. The surrounding neighborhoods, while part of the "core" or "center" or "7.2" or even "Greater Downtown" are not "Downtown."
Now, I have heard people in the suburbs say "going downtown" to refer to going to Slow's or the DIA or Eastern Market, but I think that's just a shorthand reference, not a claim that those places are part of "downtown." And it's better than when people say "going to Detroit" as if it's a totally separate place. [[personally, my preferred nomenclature is "going into the city").
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