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Yes, the terms monotypes and monoprints are often misused and misunderstood. A monotype artist myself, I am sure to make the distinction when others ask for the medium or call it a monoprint. Essentially, the differences between the two types lies in the plate surface on which the image is made. In monoprints, the unique print is pulled from a plate that already has an image incised into it. In contrast, a monotype is made on an unworked surface. The surface I work on is a plexiglass sheet.
I have done monotypes with elementary children and they loved it. In many ways, monotypes are similar to fingerpainting prints, something children are quite familiar with. There are three main types of monotypes. In one, paint/ink is applied to a clean plexiglass in a painterly manner, adding brushstrokes and then printing. Sometimes I wait until the entire image is completed before I print it, while other times (it depends on the medium) I lay the hinged paper over the surface and rub after each section is painted.
Another type is the reductive or subtractive type. Here, ink is rolled onto the clean surface until the entire plexiglass is covered with an even coat. In order to "find" the image, the printmaker must wipe away or remove ink. Qtips, rags, and even matboard scraps are excellent tools to do this.
The third type of monotype is the direct draw. Here, I can draw an image prior to getting the ink ready. In this process, ink is rolled onto the clean surface. The paper is laid on top of the inked surface, with the preliminary drawing facing up. I then begin tracing or redrawing over my pencil lines with a pen or other tool. Or, I sometimes choose to create the printed image without a preliminary sketching, remembering though that all marks will show I can't erase!! Once all the lines have been traced and the image (when I peek under the paper) looks fine, I remove the paper. Much like carbon paper, I now have a copy of my drawing! One thing, though, resist the urge to draw with your hand laying on the paper surface. The slight rubbing resulting from this pressure will show up as gray fuzziness on your image!
If you are interested in this exciting technique, I urge you to look at the book
"Monotype: Mediums and Methods for Painterly Printmaking" by Julia Ayres
"The Complete Printmaker: Techniques/Traditions/Innovations" Ross, Romano, Ross
(this book also has a great section on collographs)