Well I don't think the vocabulary is particularly important here, since they likely didn't use the words in the same way we do. Like some scripts like batard or English secretary hand were evolutions of the formal script that reduced pen strokes to be faster to write making them more cursive.

But I'm curious about the history of connected letter scripts like Italian round hand. But most of the books I've read about handwriting have been in the American tradition, and it helps they are easy to find on the Internet. Some cursory reading on the subject seems to point to it coming from Italy in the form of old Roman cursive. To my eyes old Roman cursive seems related but is too different for me to call a flowing connected letter script. This isn't surprising though since it was used to write on wax tablets.

It seems like something we would recognize in the modern world as a connected letter cursive originated in the late 15th century Italy out of italic script. But I don't speak Italian or Latin so I don't know how to find any primary sources on this.