I would argue an index, in the sense that it might be an integer, is indistinguishable from an offset. I would refer to one-based indexing as positional—1st, 2nd, 3rd, ..., ith.
Of course, it's the offset from the first element, so it's kind of a circular definition.
Generally, it's useful to distinguish offset from name sometimes. For instance, gravitational potential energy has numbers associated with different amounts, but once you choose an origin you can compute energy differences.
Another example is screen space versus screen displacement. This is the difference between affine space and a vector space. Whether the upper corner is (0,0) or (222,22) shouldn't matter as long as you are doing everything relative to some point.
In C, I argue we always use offsets. Each element of an array A is at a particular memory location, the name of the array being the first memory location, and then A[i] means take the thing at A+i. Notice that the difference p-q between two memory locations p and q is exactly the offset you put into an index expression: q[p-q] == *p.
That said, it is convenient to confuse the offset with the memory location since 1. the memory location is likely not known when writing the program 2. if it were known, it would be almost impossible to use.
Now, an anecdote: I was helping implement a QR factoring algorithm from a textbook which uses 1-based indexing in a language which uses 0-based offsets. We tried changing the bounds of the nested loops to account for the difference, but it was basically impossible to avoid off-by-one errors. So, we left the loop bounds as the textbook had them and instead indexed like A[i-1], since this i-1 is the offset from A[0], the array element labeled 1.
Of course, it's the offset from the first element, so it's kind of a circular definition.