From what I understand after reading a bunch of calligraphy books (both modern and 19th century), there two ways to use the word. Most Americans use the word cursive to mean some form of joined writing, usually something like business penmanship (or a business script). These are usually descendants from English roundhand and spencerian.

However the word can also be used to describe a scale between cursive and gothic. The more gothic a script is the more times the penman lifts their pen. The more cursive a script is, the less the pen will be lifted off the page.