Why are polynomials chosen with coefficients in $GF(2^{8})$ in Section 4.3 of NIST FIPS 197 for implementing AES?
The irreducible polynomial you showed above will create a polynomial field. (0x11b=283=100011011).
-now last nearest degree of 2 is ${2^8=256}$ hence all the operation of a field within the range of ${0,1..,255}$,

It is a Byte operation because the range ${0,1,2,...,255}$ can be represented as a byte(8 bit) and the field operation are performed and reduced to modulo ${x^8+x^4+x^3+x+1}$
Here, The statement ${\{0,1\}^8}$ means it can be a binary number of 8 bits.
But in case of normal polynomial the coefficient can be from set N,Z,R,C. which is not the case in AES.