Sorry I missed your question first time. I don't know as much about this as I'd like, but I do know a little; I am not sure that the historical truth will help much, because all these 'invented' nationalisms of the nineteenth century that took root are now so inbred into modern readings of history as to be almost impossible to root out. But let's try.
Nineteenth century ethnographers regarded the inhabitants of the eastern parts of the Ottoman Empire as direct descendants of the Assyrians in the Old Testament, and with that supreme racially-inspired confidence of the Victorians, decided to call the inhabitants 'Assyrians'. The American missionary societies, which had sent missions out to that region as early as the 1830s, originally, and more accurately, 'the mission to the Nestorians', but they later changed it to fit in with the Anglican nomenclature - which meant that by the 1880s it had become pretty much the designation used by the English-speaking world; and that is where it comes from.
The situation is complicated by the work of the Protestant evangelicals in the region, and by the Roman Catholics, as well as by the Chalcedonians, who all thought it OK to go about 'sheep stealing', with the results we can see today.
I am not sure that the Church of the East would be very willing to listen to the historical record. But if one has eyes to read, something contemporary such as Oswald Parry's Six Months in a Syrian Monastery gives a sympathetic and accurate account of the situation on the ground in the late nineteenth century. William Taylor's Antioch and Canterbury which is available from Gorgias press also gives an account of Anglican attitudes - which were, alas, entirely predictable.
I shall do a little more digging and get back on this, if I might?
In this is love, not that we loved God, but that He loved us and sent His Son to be the propitiation for our sins. (1 John 4:10)