Re: My cordless landline doesn't, and that's a phone.
Can I be a pedant for just a moment? I know that usage trumps all else but; the word 'mobile' accurately means 'movable by vehicle'. Think of a mobile library. A mobile home. A mobile disco. None of those things will fit comfortably in your pocket.
The term 'mobile phone' was the US equivalent of the British 'car phone' - a device that predated cellular telephones by maybe 20 years (google the GPO's Radiophone / System 4 service). We even have a national chain named "carphone warehouse".
When we say 'mobile phone' in modern usage, we really mean 'transportable phone' if we're talking about the specific device type, or maybe "cellular phone" if we want a catch all.