Two definitions of the Reed-Muller code
They are the same.


In the first definition, we identify $\mathbb{F}_2$ with elements $\{0,1\}$, with $+$ defined to be addition modulo 2 and $\times$ defined to be multiplication modulo 2.
In the second definition, we identify $\mathbb{F}_2$ with elements $\{1,-1\}$ and $+$ defined to be multiplication modulo 2 and $\times$ defined as $x \times y = \max(x,y)$.


Both representations of $\mathbb{F}_2$ are isomorphic.

Degree is defined as usual for the polynomial ring $\mathbb{F}_2[x_1,\dots,x_n]$; see your favorite math textbook for the formal definition.  Do note that $f$ is a multivariate polynomials, not a univariate polynomial.



The "integer coefficients of magnitude at most $d$" is weird and doesn't belong.  The definition would make more sense with that phrase omitted.  Pedantically speaking, that phrase makes no sense, as a polynomial over $\mathbb{F}_2$ by definition has coefficients in $\mathbb{F}_2$, so it makes no sense to speak of them as "integers", and they don't have a magnitude.  (Alternatively, if we interpret it loosely/informally, that phrase becomes redundant and dead and ultimately has no effect.)