Since you are asking about a place name, and especially because you are looking at a record on Ancestry, I think it's important to step back and get a better understanding of what you're looking at.
Ancestry says this list of births is from Cane Creek Monthly Meeting in Alamance County, North Carolina, from a volume titled: Marriages, Vol 01, 1756-1840.
What's wrong with this picture?
If you go to the Atlas of Historical County Boundaries at the Newberry Library, and look at the Individual County Chronologies for North Carolina, you'll see that the entry for ALAMANCE county says:
19 Apr 1849
ALAMANCE created from ORANGE. (N.C. Sess. Laws 1848-1849, ch. 14, sec. 1/pp. 39-40, and ch. 15, secs. 22-25, 28/pp. 40-48; N.C. Sess. Laws 1852-1853, ch. 22, secs. 1, 3/pp. 71-72)
So the Cane Creek Monthly Meeting, which may be in Alamance County now, couldn't have been in Alamance County at the time this meeting closed out the book, assuming the last entry is indeed from 1840. Alamance County didn't exist yet.
On the record page, Ancestry offers you a link to Search for this monthly meeting in the 'Quaker Monthly Meetings Index' but their link only searches for other references to Cane Creek Monthly meeting in their own database (which is useful if you're looking for references to people who left Cane Creek Monthly meeting, but that's not what we want right now).
Searching Quakermeetings.com for Cane Creek in North Carolina gives us much useful information about the meeting, starting with this:
State or Province: NORTH CAROLINA
County: ALAMANCE, WAS ORANGE, + CHATHAM, GUILFORD
Physical location: SNOW CAMP 27349, 719 W. GREENSBORO-CHAPEL HILL RD.
Latitude: 35.89925° N. Longitude: 079.41053° W.
Date granted: 1751/10/31
Date of first meeting: 1751/12/07
Current Status: ACTIVE
Before and After: FROM DUNNS CREEK (CARVERS CREEK) MM. PM 1760BEF.
The listing goes on to say what the Yearly Meeting is, where the original records are kept, known extant records, what local histories mention this monthly meeting, and lists the associated meetings -- all useful information if you are tracing Quaker Families.
Where records are kept: GUILFORD COLLEGE mf LDS 371251, SWARTHMORE
The film notes for film number 371251 in the FamilySearch catalog read:
The catalog notes say:
Microfilm of originals in the Guilford College in North Carolina.
Includes minutes, births, deaths, marriages for 1755-1840 and other miscellaneous information.
However, note that Ancestry doesn't cite FamilySearch as its source for this collection of Quaker Meetings. Ancestry says the records come from:
North Carolina Yearly Meeting Minutes. Hege Friends Historical Library, Guilford College, Greensboro, North Carolina.
Scrolling back to the start of this group of images, the first image after the title card seems to be the endpapers of the book, on which someone has written in modern handwriting:
Western Quarter Cane Creek M. M.
Records, Vol I
Presumably Ancestry used this information when making their waypoint label. All of this suggests that Ancestry could have made an independent imaging rather than a copy of the film at FamilySearch, but it's difficult to know without comparing the images to the film.
In conclusion -- given the history of Cane Creek Monthly Meeting, and the county boundary history for the area, it seems likely that what you've read as "Orring" does refer to Orange County.
For ORANGE the first entry in the county chronology reads:
31 Mar 1752
ORANGE created from BLADEN, GRANVILLE, and JOHNSTON; western boundary limit was indefinite. (N.C. Recs., ch. 6, secs. 1-2/23:383-384)
This predates the individual record you linked to for Elizabeth Wright that referred to her birth dated 15 Twelfth 1756, so Orange County was in existence at that time.