That is a terrible definition of a 'smart phone'. My last phone was an Alcatel black berry knockoff thing. Now blackberry could make an argument for being an entry level smart phone even without a touch screen... but the Alcatel phone I had was decidedly not 'smart' by any definition of the word.
To me a smart phone must have the following:
1) a decent web experience where most pages work, and are displayed correctly
2) work network capability, meaning it can plug into productivity software such as exchange/outlook or sync data files
3) a rich media player expierence where most formats are supported and display correctly at a decent frame rate
4) a rich communications suite which includes phone, sms, chat clients, email and forums
5) a comprehensive contact list which the phone is aware of, and can attribute to your various forms of communication to each form of communication that the phone supports.
Obviously the big 3 of Android, iOS and WP7/8 have more features than this, but I think that any phone OS that has at least these things could be considered a smart phone no matter the form factor or touch capabilities. While my old Alcatel had decent hardware which was near identical to a BB device, whatever OS they used definitely disqualified it for any 'smart' label that could be attributed to it.