Personally, I don't get the focus on joined up writing, but I'm not opposed to people teaching how to write in an efficient legible manner.

The problem with the word cursive is as you say. Some people mean "handwriting" and some mean "calligraphy".

In the US, cursive is the last example from the first image. I think we should primarily teach the first three from the first image because they match the typographic style used everywhere in our society.

The second image, which claims to be how the UK does it, is acceptable because it's basically print but with standardized "pen drag". I don't see the point anymore to telling people not to lift the pen, but I don't think it's backwards or silly.

A good litmus test came to me while looking for those images: in a good number of handwriting examples people write with joined letters that they can't do cursive at all, and actually can't remember most of the letters.
I think if it gets to the point where you need to remember a second set of letters, and not a method for writing one set of letters, we shouldn't make it a core requirement.