(I love persimmons too.) Overall this doesn't seem to have much similarity to persimmon other than sweet taste and aroma (which is true of many plants). And if I remember correctly, persimmon originated in China and wasn't known in the west until after medieval times (I'm going by memory, I'll have to check on this).
As in the case of cyclamen, sometimes the name changes when a new plant is introduced. For centuries, one of the predominant names for Cyclamen was earth apple (it was also called cyclaminos, orbicularis, and pig's bread), then the potato was introduced to Europe and it became the earth apple (pomme de terre).
Assuming the name in the Trinity herbal has any validity (I don't know whether it does), maybe "kaki" in Greek referred to something different before the persimmon was introduced.
As in the case of cyclamen, sometimes the name changes when a new plant is introduced. For centuries, one of the predominant names for Cyclamen was earth apple (it was also called cyclaminos, orbicularis, and pig's bread), then the potato was introduced to Europe and it became the earth apple (pomme de terre).
Assuming the name in the Trinity herbal has any validity (I don't know whether it does), maybe "kaki" in Greek referred to something different before the persimmon was introduced.