That quotation from Apple's website doesn't tell the whole story......
Verizon and Sprint (both giant greedy dickwad carriers BTW) refuse to allow a person to use any tablet on their network that was not originally designated as a "Verizon" device (or sprint).
Its 1000% BS, and I hope Verizon Corp takes a tumble like Tyco or Enron.
BUT if you want to use the device on Verizon EVER, you better buy a Verizon version upfront. AT&T and Tmobile would rather have your business, and they don't care who the tablet was originally labeled to be used with. All iPad Air's contain the correct LTE radios to work on any of the big 4 networks.
P.S. Verizon sucks @$$.
Disclosure: I have Verizon as my phone carrier (due to the nature of my job). I have Tmobile as my iPad carrier.