The first alphabet below is true Cursive. They are, with colors reversed, the couplets that ran the breadth of your grade-school classroom, right above the blackboard.
This second alphabet may look "cursive", but it isn't. It's a writing system called the Palmer method (the Wikipedia link is here), and has some notable differences - the capital "T", "F", "P", and "R", for instance, and the eerie-looking small "r".
I remember the "Palmers" because my first name starts with a "T", and I was given the option of using the cursive one or Palmer's. I chose the cursive, although as horrible as my handwriting was (and is), you couldn't really tell the difference. The Palmet "T" looks dorky - like a stiletto heel with a beret.
As mentioned earlier, cursive has pretty much bit the dust. Most of us now use a hybrid system mixing some cursive with block letters. The "T" in my signature is block - one line down, one line across. Palmer's system was pretty much dead when I was trying to write legibly back in the late 50's.
It should be noted that there are variations of the Palmer letters (they even show two capital "E"s above). That "F" is ridiculous; I was taught to make the Palmer "T" and just put a line through it, a la the Cursive.
Enough about cursive. I still don't know why I was dreaming about it the other night. And I don't really care. I have re-learned how to make the cursive small "v", and that allows me to sleep comfortably at night.