Exactly my point, cursive refers to that exact style of joined-up writing. It is not a word which refers to any and all joined-up writing. If you ask almost any American what cursive is, they’ll point to the style as shown in the first image.

If you ask almost any American what “joined-up writing” is, they’ll point to the style in the second image. Because if you were referring to Cursive, you would have said cursive, not joined-up writing.

If you avoid using the very common specific term for something in favor of a less common, more general term, people will assume you’re not referring to the more specific thing, because you’ve gone out of your way to not use the specific term.