Now I understand what you're saying, although I think that it is a pretty high standard, that would exclude not only most calligraphy, but also most writing styles, as being classified as writing.
Not at all. It isn't just the pen lifts, but also the speed at which it can be written efficiently.
Consider a writing style one that is straight forward and efficient. For example the block lettering taught engineers, or the script taught to Navy radiomen taking morse over the radio -- all capital letters, angular, and definitely not a flowing cursive style. You do lift the pen at times, but it is fast, efficient, and makes for a legible hand even at high speed. Architectural script is not connected, but it is all monoline and efficient. This is a writing style, not a calligraphy style that's purely ornamental.
How about Spencerian? Lots of that is meant as a practical hand, the ornamental offshoots of the basic style being concentrated on motions that are almost natural movements of the pen -- like the bottom of a P or the swell on the lower half of a capital stem that all come with just increased pressure on a downstroke -- but the lower cased alphabet can be written entirely monoline and at a good speed and still be recognizable as Spencerian. This is the basis of my daily hand, in fact.
Maybe look at it from a different angle. The point isn't JUST the pen lifts, it's the speed and formation of letters. A script you can make taking notes or writing a letter to a friend, where you're thinking about what you are going to say as opposed to the letters themselves, is more normal writing. When lower cased letters require slow, careful drawing of each stroke in a complex fashion, then you are on the other side of the line. I cannot do engrosser's script as a monoline at a reasonable speed, like regular handwriting, and still recognize it as engrosser's script.
I think that's what's meant, at least.