The starter can be kept in either liquid form or a more solid form. Since soft starter doughs favor the production of lactic acids and stiffer doughs favor the production of acetic acids, bakers take advantage of this fact to achieve the type and intensity of flavors in the finished product, such as bread or pizza crust. Also, some starters are naturally more sour than others, an example of this being the Ischia starter which is naturally more sour than the Camaldoli starter, which yields a ratio of lactic acid to acetic acid of about 3:1. It also acts faster than the Ischia starter. Acids also affect enzyme performance and can be used to extend the fermentation time of a dough. However, you don't want to have a too-acidic environment since that can affect yeast fermentation. So, the selection of the type of starter and its form and quantity will be major factors in the development of the dough during fermentation and the flavors in the finished crust. This is a somewhat oversimplified explanation of very complicated subject but I think it addresses the question you raised about the form of the starter.
I believe the piece of dough in a cloth bag may be a dried form of a starter. Alaskan prospectors used this technique (and other similar approaches) to protect and carry their starters from camp to camp. Usually the starters so preserved were activated the day before intended use.