why does my iphone 5s have a 1440 mAh battery?

Battery is shot in this IPhone, so I need to get a new one. I purchased a battery which states on every website I go to that its a 1560 mAh battery for the 5s. I open up my phone to put the battery in and it wont fit!! the connection is slightly larger so it wont snap in. I look at both the batteries and I see the one I purchased is indeed 1560 mAh, but my OEM battery is 1440 mAh. the connections are obviously different as well! so why do I have an Iphone 5 battery in my Iphone 5s? do I need to purchase an Iphone 5 battery to make this work? I am absolutely positive this is an Iphone 5s because it has the fingerprint reader. this doesn't make sense!!