They are water heaters, at least as far as plumbers are concerned.
http://www.plumbingsupply.com/w-x-y-z.html
http://www.aceee.org/consumerguide/topwater.htm
http://www.austinplumbingsupply.com/austinplumbingsupply/Water%20Heater/Water%20Heaters.htm
http://www.lowes.com/lowes/lkn?action=productList&N=0&Ntk=i_products&Ntt=water%20heater
Working as a materials buyer for a plumbing shop for a couple of years I have heard both and I think either is equally correct.
When you first install a unit the water inside is cold. It acts as a Water Heater and a Cold Water Heater.
After a couple of hours the internal water is over 110 F and the heating units will come on every so often (as the thermostat dictates) to reheat the Hot Water. At this time it is decidedly a Hot Water Heater and a Water Heater.
While it is true that it originally is a Water Heater, for most of it’s functioning life it is indeed heating (or reheating) water that is hotter than ambient temp.
Additionally, every invoice and work order I have seen abbreviates it as a HWH.
Of course, YM(and HWH)MV. 
I had mine replaced in June. The invoice says, “WATER HEATER.” Yes, it’s redundant, and maybe dunderheaded, to say “hot water heater,” but that’s how most people say it.
I’ll make a fuss about some points of grammar and usage, but this isn’t one of them.
Your wife is wrong. If you both want to win get a tankless water heater…