The formula x=-x is valid for all fields of order 2^k, not just the field of order 2.
-1 mod 4 = 3 != 1, so x = -x doesn’t seem true in the sense above outside mod 2
Yes, but the field of order 4 is not the same thing as the integers modulo 4. The field of order 4 has the four elements 0,1,x,x+1, where x satisfies the equation x^2 + x + 1 = 0 and 2=0. For more details see https://en.wikipedia.org/wiki/Finite_field#Field_with_four_elements
The formula x=-x is valid for all fields of order 2^k, not just the field of order 2.
-1 mod 4 = 3 != 1, so x = -x doesn’t seem true in the sense above outside mod 2
Yes, but the field of order 4 is not the same thing as the integers modulo 4. The field of order 4 has the four elements 0,1,x,x+1, where x satisfies the equation x^2 + x + 1 = 0 and 2=0. For more details see https://en.wikipedia.org/wiki/Finite_field#Field_with_four_elements