Even if it doesn’t explain the difference in rendering between Dorico and InDesign, the complete version of Minion Pro (through CC) will probably include a weight that will produce output from Dorico that very closely matches InDesign.
You might have noticed that the complete version comes with 65 fonts! They are organized into groups at several “optical weights.” Each optical weight includes all of the normal font variants (regular, italic, bold, etc.) but is designed to maximize legibility within a particular range of text sizes. The optical weights are:
Caption — best for small type.
Regular — best for type at typical paragraph sizes.
Subhead — best for type a bit larger than the standard paragraph.
Display — best for especially large type, like titles.
Each optical weight includes regular, bold, italic, etc. variants, so you’ll be able to find fonts like “semibold subhead” and “bold italic display.” The differences between optical weights are meant to be subtle, but one of the most important aspects is that smaller optical weights feature much less contrast between the thickest and thinnest parts of the glyphs than do display weights.
Again, I’m not confident that the difference in output that you’re seeing results from some behind-the-scenes use of different optical weights, but I do think that the complete collection of weights will give you enough flexibility to achieve the desired result. I’ve attached a screenshot of a PDF produced in Dorico with your text at several optical weights for comparison. Hope that helps!