I agree that the Piedmont is as you stated, though Oxford pulls from the Italian: Piemonte. However, I disagree that the Foothills are the same thing. Though the Piedmont may stretch from the base of the mountains to the fall line/coastal plains, The Foothills is a specific area: a distinct lower part of a mountain(s) - singular; a hill that leads up to a mountain. I will concede that the Foothills are a part of the Piedmont region, but would argue that it is the western most part of the Piedmont. (Squares can be rectangles, but rectangles can't be squares.)
A very similar argument could be made out east for the Sandhills in the Coastal Plains. There would also be some people out there that would argue that there is a Tidewater Region that should be separated from the "Inner" Coastal Plains. But yes, there are 3 main geographical areas, historically.
Geologically speaking, I would say the line for the foothills falls between the established western line that goes through the western half of Polk, Rutherford, Burke, Caldwell, Wilkes, and Surry counties and my perceived eastern line going through half of Cleveland, Western most part of Lincoln, half of Catawba, northwestern most part of Iredell, most of northern Forsyth and most of west and north Rockingham.
For clarity, I'll take your reply one paragraph at a time, starting here: You created your own disagreement. Originally, you had tenderly asked, "Is there an eastern foothills?" My answer was etymologically and geographically (I use these two terms as qualifiers), yes... I would cite the definition of the word Piedmont. Either dictionary should bear me out on this.
This is the excerpt from my answer "To answer your question, though, I say yes, there is geographically and etymologically speaking an eastern foothills, namely those Piedmont counties just west of the Fall Line/ NC Costal Plain.
Granted, it's flatter than what we think of foothills as being but generally some rise in elevation if someone is going from counties east of the fall line to. counties west of it... You suggested that I said it was the same thing. That was not said...You headed up the first paragraph by citing the difference between squares and rectangles and how one couldn't be the other but as you know, even a rectangle has four sides.
In your second paragraph, you made reference to The Sandhills and Tidewater areas. I suppose this was mentioned to point out that specific areas have certain labels, but I'd have to refer back to your original question, "Is there an eastern foothills", and again I would argue that based on the meaning of the word Piedmont, and a gradual but slight, and sometimes intermittent rise in elevation, yes, it could be argued that there is an eastern foothills based on this.
On your last paragraph, yes, I think that's fair, the line you laid out, but since you mentioned
geology, let me point out something else that could go to my argument of an eastern foothills. The NC Fall Line which ends the piedmont and begins the coastal plain, is not simply an imaginary line, but a drop-off, in which the land begins its descent towards the coast. There is a difference in elevation, granted that it could be slight in some places, but generally, there is a decline in elevation. This rise is generally west but it can be northward, too, considering your rectangle which is a similar shape of the NC Piedmont.
Early on in my answer to your question "Is there an eastern foothills?" it was mentioned that I had never heard of it, but for the sake of technicality, yes, I do believe an argument could be made for it considering what two qualifiers I used.