You’re right. I misunderstood my first source.

> After the creation of a special cavalry unit, the peoples of the ancient Near East and China made some innovations to the equine equipment in order to control their horses during fighting. The “martingale” collar was probably an innovation of the Near East, and stirrups were invented in ancient China.

On my first read, that sounded to me like these happened concurrently. Upon closer reading, the paper actually indicates China had cavalry 600 or more years before the stirrup. Thank you!