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1. The Original iPad mini. (Released 2 November 2012) This model doesn't have the extra noise cancellation microphone on the back. And, of course, it doesn't have Touch ID. Just turn it over and look at the top of the back to see that there is no extra microphone. You're done.
iPad mini. No noise cancellation microphone at top middle.
2. iPad mini 2. (Released 12 November 2013) This iPad introduced the Retina display, but for many, it's hard to visually see the difference right off hand. It's faster to just turn it over and look at the back, at the top. You'll see the tiny noise cancellation microphone.
iPad mini 2. Has the additional microphone marked with a red arrow at top.
3. iPad mini 3. (Released 24 October 2014) This iPad has a Retina display, the noise cancellation microphone and adds Touch ID. That's easy to distinguish thanks to the metal ring around the home button. If you see that, you instantly know it must be an iPad mini 3.
Left: Normal (old) Home button. Right: Touch ID Home button on iPad mini 3.
(Note the edges of a plastic screen protector on the left image.)
It sure would be nice if Apple were to include not only the model number but the name of the product in Settings > General > About, but that's not Apple's way.