I think I misunderstood something simple but not sure what.
According to https://en.wikipedia.org/wiki/Coxeter_element,
The invariants of the Coxeter group acting on polynomials form a polynomial algebra whose generators are the fundamental invariants; their degrees are given in the table above
The Coxeter group $A_{n}$ is a group generated by reflections in the hyperplans defined by $e_{i} - e_{i+1} \in \mathbb{C}^{n+1}$ where $\{e_i\}$ is a orthonormal basis of $\mathbb{C}^{n+1}$. So $A_{n}$ permutes coordinates of $\mathbb{C}^{n+1}$ and it is in fact isomorphic to the symmetric group $S_{n+1}$. Therefore the $A_{n}$-invariant polynomials of $\mathbb{C}[X_1,\dotsc,X_{n+1}]$ are elementary symmetric polynomials \begin{equation} Y_1 = X_1X_2\dotsm X_{n+1}, \qquad \dotsc \qquad Y_{n+1} = X_1 + X_2 + \dotsb + X_{n+1}. \end{equation} So the degrees of fundamental invariants should be $n+1,n,\dotsc,2,1$ but the same wikipedia page (and every other sources I checked) will say these degrees are $n+1,n,\dotsc,2$. Apparently we will always restrict the action of $A_n$ to the subspace $X_1 + \dotsb + X_{n+1} = 0$ in $\mathbb{C}^{n+1}$ and ignore $Y_{n+1}$ but why is that? Is $Y_{n+1}$ not a valid invariant polynomial of $A_n$?
The degrees of the fundamental invariants are not invariants of the Coxeter group but rather depend upon the particular representation. More specifically, the degrees are defined for any finite subgroup $G \subseteq \mathrm{GL}(V)$ with the property that the invariant subalgebra $\mathrm{Sym}(V^*)^G$ is a polynomial ring: these are the degrees of a system of homogeneous generators. If the vector space $V$ is defined over a field of characteristic $0$, the groups with this property are precisely the reflection groups; if the field is the real numbers then each reflection group is isomorphic to a finite Coxeter group and each finite Coxeter group occurs in this way.
The point for you is that the embedding $G \subseteq \mathrm{GL}(V)$ must be specified before you can make sense of the invariants. For the symmetric group $S_n=W(A_{n-1})$, it is customary to refer to its permutation and its reflection representation, of dimension $n$ and $n-1$, respectively. Each comes with its own set of degrees.