Simplest solution
My first thought would be to create dashboards or landing pages, that way you could have both a Chart and a Table together, the table being the one people export and the one you can control the columns.
More complex
Insights can be sourced directly from different data pages BUT only in Design Time (App Studio > Explore Data) and only on non-queryable data pages.
I’m not really a fan here, as you’d be hand crafting a lot of reports and data pages just to support this requirement. The earlier option above seems more logical.
It is covered a little more in Some insights into Insights (and Explore Data) - #27 by MarcCheong