After that, the iPhone was clearly something qualitatively different, so people (or rather, journalists, phone stores and the like) started calling the old style “feature phones”.
I’d say it’s now misleading and even incorrect to call something a “smartphone” if it doesn’t have a full-size touchscreen.
You don't misrecall. Even phones without a full keyboard like Nokia's N-series (e.g. the very popular N-70 https://en.wikipedia.org/wiki/Nokia_N70) were considered smartphones.
Exactly, "feature phones" were called that because of how the cell phone companies had to try to sell previous-generation phones in the same market as the iPhone. When other cell phone companies saw the iPhone, they realized the only way to present their phones as competitive was to frame the iPhone as a bundle of features (music player, web browser, games) and say hey, that's not so special, our phones do that stuff too. So you had flip or candy-bar phones that had an app for playing music, a terrible, crippled mobile web browser, maybe email, maybe a weather app, and some built-in games.
Calling these phones feature phones is a pretty awful knock, in other words. It means they aren't comparable to smart phones.
All deserve to be called smart phones.
"Feature phones" were at one point the high end of a carrier's phone lineup, as they offered features, such as music playback, a camera, or WAP internet access, that a basic phone did not. Usually they could only run very limited applications, written in things like Java ME, that were available for sale through your cell carrier. When the iPhone was released, feature phones became the new low end. (Pre-iPhone smartphones were quite rare, usually being carried by on the go execs and the like.)
It was, of course, absolutely bloody awful and didn't have even enough RAM to do the most basic things. But it was 3G and I had an unlimited plan that I tried to make the most of by watching videos etc on the go. I even used to use it as an MP3 player.
If you're looking for a word to describe phones whose interface consists of a touchscreen with icons and gestures, I'd suggest "phablet".
When the iPhone launched, people called it a smartphone even though you couldn't install apps. Multitasking was also years away.
"Smartphone" has never meant anything really. Just marketing.
By your definition the original iPhone (3.5” screen) was a phablet.