We do know that even over a thousand years ago, speakers of Old English were still calling these kinds of fruits berries, such as strawberries and blackberries (although pronunciation differed somewhat, of course). A word for strawberry as "earth berry" is even reconstructed for the proto Germanic language around 1500 to 2500 years ago. Beyond that, it becomes difficult to trace the word berry any further.

The Botanical sense of the word berry seems to come largely from at earliest the 1500s, from the writings of Caesalpinus, although the definitions were inconsistent and later writings on the matter constantly redefined things and added new terms. Although, largely, these writings all used Latinate terms for their botanical concepts, such as bacca (the closest to the modern botanical berry), and also words like pomum (pome/pomme), drupe, etc. for the other categories of fruit.

So, somewhere since all of that, some English-speaking botanist decided it would be a good idea to use the word berry to describe this concept of a bacca (even though berries had been used for distinctly different things from what that concept described), and now we end up in our current silly predicament where strawberries aren't berries but pumpkins are.

I'd propose we call botanical berries "bayes" or "bayfruit", the word bay/baye being an alternate word for berry that ultimately derived from the Latin word bacca, via Old French.