It depends on the game, really, but yes, you'd normally need to design for the screen itself.
DPI isn't the term you are looking for but PPI, which is a measurement of screen density. Hopefully, the higher-density screens on other devices will adopt the iOS model where the screen is double-density (retina display) but still uses units of measurement that are akin to the low-density devices. In other words, an iPhone screen is 460x320 'pt's even though the iPhone 4 has 960x640 actual pixels.
But...not all devices do that. For instance, there's a handful of BlackBerry devices that are higher density, but you can't easily accommodate them without doing specific device detection.
As for your second question: "If you had to make 2 designs for low and high resolution, which ones would they be?" it would be entirely dependent on the platforms you are targeting.
If you are aiming for ios, then that's easy. Design an iPad layout and an iPhone layout. For the iPhone version, you'd create two sets of image assets (standard and retina).