But Topsie, there were "mobile" phones around long before there were "convenient" mobile phones.
There still are a few accessible "relay networks" around that are used both by commercial comunications and by amateur radio enthusiasts that are "mobile phone" networks but are NOT "cellular phone networks."
It's the development of networks of antennas/antennae/pointythings each covering an area within a "cell," and the method of linking the "cells" together in a way that let's someone move from one cell to another without interrupting the communication, that defines a cellular network, and the rather specialized cellular phones that use that particular method of communication.