So don't set running type in "exotic" faces; use Minion for display type and subheds, and use a compatible standard face for the body.

The main reason to use Minion is that it’s an awesome text face which ends up taking up a bit less space than other faces at a similar size without sacrificing readability (I used it for almost all my papers the last couple years of college, for example). As a display face, it’s not terribly exciting (especially since it doesn’t seem that the fonts designed for display size are included here; so these Minion fonts are intended for 10–12 pt. on paper).

Weird, I'd think that they would all use the same font rendering provided by the OS.

Well, Chrome actually appears to use font rendering from Windows (it looks the same as IE8), and Indesign has a good reason not to.

IE9 shot would be interesting to see for comparison. Based on what IE people were saying over at Typophile forums, they made a targeted effort to dramatically improve the rendering quality of IE font rasterizer.