Exporting for web and mobile has very simple rules. There are only two:
- convert to sRGB
- embed the profile
That's it. That covers the maximum amount of possible scenarios and viewing conditions. There's nothing more you can do.
In Save For Web, you need to set Preview to "Use Document Profile". That's the color managed and correct version, and the one you will see in Photoshop. "Monitor Color" displays the image without any color management at all, which is useless as it doesn't tell you anything about how others see it. The other two settings are hybrid settings that are just confusing and equally useless. In Export this is tied to the embed profile checkbox.
Phones have no color management. Anything can happen there, it's out of your hands. Many of them have various types of wide gamut displays, which means they always display oversaturated. Nothing you can do.