Well, don’t I feel stupid! The Rumi referred to in my Arabic workbook is Ibn al-Rumi, a fact which I completely ignored as I was writing my earlier post. That would explain why they refer to him as being of Byzantine background, because Ibn al-Rumi was of Greek descent and did live in the 9th century.
In fairness, this Ibn al-Rumi was, as I have discovered, a native of Baghdad and has a rather indirect connection to Rum in any case. This makes the claim about a “Byzantine background” for him a little odd. Next time, I’ll be a bit slower to jump to conclusions. Such are the perils of the blog.