What formula is applied to RGB channels with Image>Mode>Greyscale (discard color information)

According to Thomas Knoll (through Bruce Fraser), the original Convert to Grayscale using the Mode Change menu command uses the following weighting: 30 percent red, 59 percent green, and 11 percent blue. 

Convert to Grayscale using a profile is different of course and as outlined, depends on the color settings and what you selected for Gray there (gamma, dot gain or a profile you select). So in these more modern workflows, YMMV. 

Image > Mode > Grayscale converts to whatever grayscale profile you have set as working gray. It's a standard profile conversion just like any other profile conversion, going through Lab or XYZ as Profile Connection Space.

This not the same as the Luminosity blend mode in RGB. Here there is no actual profile conversion, and a formula is used to mix the three channels. It has to work that way to give reasonably consistent results no matter what RGB color space you're in.

Proper grayscale is subject to standard color management just like RGB or CMYK. Obviously there is no "color" involved, but the different profiles have very different tone response curves.

The problem is that very few applications outside Photoshop have proper grayscale color management. This is why a grayscale image may look completely different in other applications. Without color management, you need to match source and destination, and so you need to pick a grayscale profile in Photoshop that matches the actual destination. For instance, if it's for screen viewing, you would use Gray Gamma 2.2.