I'd say a creek, run, and branch could all be the same size, from small to large, around Dinwiddie. Also, rills would be the same. However, I would say that creeks and runs would go from small-sized to very large, whereas a branches and rills would tend to remain on the small to medium size.

bartleby indicates regional variations in usage in the eastern US: Terms for "a small, fast-flowing stream" vary throughout the eastern United States especially. Speakers in the eastern part of the Lower North (including Virginia, West Virginia, Delaware, Maryland, and southern Pennsylvania) use the word run. Speakers in the Hudson Valley and Catskills, the Dutch settlement areas of New York State, may call such a stream a kill. Brook has come to be used throughout the Northeast. Southerners refer to a branch, and throughout the northern United States the term is crick, a variant of creek.
http://www.bartleby.com/61/59/R0345900.html, at end

Around here, branch refers not to the size of a stream but to its configuration. When a stream (whether a river or a creek) splits into two roughly-equal sub-streams as one travels upstream, each is called a branch. Thus, one has the north and south branches of the Chicago River, and similarly, the north and soutch pranches of many creeks.

When a brach similarly splits, each part (or the smaller part) is called a fork.

I'd think a river is distinguished from all of these term in that it must be big enough to navigate, at least in a substanital part of its length of a substantial part of the year. But I can't find any source to support that distinction.