While he definitely comes across as clueless, is "smartphone" really an adequate name?
Yes. It is the standard, universal term.
There are "feature phones," which may offer one or two modern conveniences (like a camera or GPS) but lacks access to more robust software-based features, high-speed data or a touchscreen. If there's a physical numerical keypad, you're dealing with a feature phone. Then there are "smartphones," which typically run iOS or Android and offer an expanded app-based feature set and 4G LTE data speed. Most smartphones lack physical key-based input; the few that do typically have QWERTY keyboards instead of numerical keypads.
"Cellphone" is a much more generic term, though its age leads people to associate it more with feature phones. If you're not sure whether the phone you own is a smartphone, then you almost certainly don't own a smartphone.