I agree with your 1+1+1 = 1. But how do you manipulate other elements than 0 and 1 in GF(256)? can you write an element in GF(256) that is not 0 or 1? (it seems that there is no other way than the form using the root of the irreducible polynomial, am I correct?)
I thought the integer representation could also be useful (and quite common), hence my 2 conversion functions, but I do not know if this was already provided by Nemos?