Confusion about Kindle and Wifi may possibly arise because of the variety of devices that you can use to read Kindle format books from Amazon.
I have the Kindle software installed on 3 different devices, and all three can (and do) download the books using Wifi. None of them are actual Kindle machines though (which use 3G, as mentioned above).
One is an iTouch running the Kindle app - so I use wifi exclusively to get the books onto that. Another is a Tablet computer running Kindle for Windows, and I use Wifi for that one too. The third device is a desktop iMac which could use either wifi or the cable connection to the modem/router.
All three are authorised to my account at the Amazon site so when I've bought a book I can download it to either one or all of them.
I hope that doesn't confuse things.
Cheers,
Chris