Another key to flexibility, in addition to proper sizing/starching of the proper weight and weave of fabric, is Thin Layers of Paint.
For those who opt for modern latex paint, either out of ease of use, concerns of safety, or simple farbery, the resulting product will be much stiffer than a comparable oil based paint process.
Use does make for a very flexible product--and eventually the item will break down. Over a decade ago, I made a number of civilian patterned floor cloths, some on heavy canvas, others on lightweight duck. Today they are all easily foldable, with the lightweight stuff now well worn from many events and begining to tear easily.
Contributing to that problem is my own sense of 'overkill'--about three years ago, I decided to recoat some pieces where wear had made them less water resistant (I used them on the floor in my home as well as at events) . With my intent to make them really water resistant, I coated both sides with linseed oil. That was a mistake, and has contributed to the breakdown of the cloth.
You are indeed correct. The term "starch" is a generic term for any substance used to fill the pores of the cloth prior to treatment. I've seen no fewer than a half dozen period receipts (recipes) for cloth treatment, but the word starch has been seen in the writings contemporary writings more often than not. What starch was made of is probably truly as you mention a regional thing. I've found no matter what you use to treat the cloth, it is a vital part of the process
Throw away that roller! Rolling the paint on pushes the paint into the fabric further making it stiffer. Also get some good cotton duck or drill and don't use the drop cloth. That stuff is loosly woven which is going to cause it to hold more paint and be more stiff. Brush on thin layers by thinning out your paint and making several coats. You'll be more pleased with the results.