I too would guess Beck Creek to be around the Hildebrand-Beck Mill, however, there was a Becks Creek Post Office in Cherokee Country near Tahlequah in the 1880s.
I found Dick Oldfield, Corporal, in Company I, 2nd Indian Home Guard (Union). I presume this to be Richard, the brother of Capt Dumplin O'Field and Lt Johnson O'Field of Co. F (2nd Org) 1st CMV (Cherokee Mounted Volunteers - Confederate).
Johnson O'Field should not be confused with Johnson Fields who was in Co. H (2nd Org) 1st CMV (and my ggg uncle). However, Johnson Fields is shown as being in Co. D, (1st Org) which was Dumplin O'Field's original company. One might speculate that the Johnson Fields shown in Co. D was actually Johnson O'Field. The captain of Co. D. (1st Org) was James Madison Bell, Watie's brother-in-law and neighbor, who was later colonel of the regiment. Watie and Bell lived near what became the Progress Community. Do you know the proximity of the O'fields to the Waties and Bells on the Grand?
In the rosters of the CMV, the O'Field brothers are shown as Dumplin O. Fields and Johnson O. Fields. Also listed is is "Johnson Fields" of the very large Fields family who were primarily living where the present Dodge Community is located -- about 8 miles east of Grove.
As you know, on the Drennen Roll of 1852, these brothers are listed as Dumplin, Johnson, and Richard Oldfields.