I believe that the minimality of these degrees for representations over the field of order $2$ was proved originally in the paper

A. Wagner. The faithful linear representations of least degree of $S_n$ and $A_n$ over a field of characteristic $2$. Math. Zeit, 151:127–137, 1976.

For $S_n$, the minimal nontrivial irreducible representation has dimension $n-1$ or $n-2$ for all $n$, as you conjectured. For $A_n$, there are a couple of exceptions: $A_7$ and $A_8$ have representations of degree $4$.

I believe that the minimality of these degrees for representations over the field of order $2$ was proved originally in the paper

A. Wagner. The faithful linear representations of least degree of $S_n$ and $A_n$ over a field of characteristic $2$. Math. Zeit, 151:127–137, 1976; DOI: 10.1007/BF01241824.

For $S_n$, the minimal nontrivial irreducible representation has dimension $n-1$ or $n-2$ for all $n$, as you conjectured. For $A_n$, there are a couple of exceptions: $A_7$ and $A_8$ have representations of degree $4$.

I believe that the minimality of these degrees for representations over the field of order $2$ was proved originally in the paper

A. Wagner. The faithful linear representations of least degree of $S_n$ and $A_n$ ver a field of characteristic $2$. Math. Zeit, 151:127–137, 1976. o

I believe that the minimality of these degrees for representations over the field of order $2$ was proved originally in the paper

A. Wagner. The faithful linear representations of least degree of $S_n$ and $A_n$ over a field of characteristic $2$. Math. Zeit, 151:127–137, 1976.

For $S_n$, the minimal nontrivial irreducible representation has dimension $n-1$ or $n-2$ for all $n$, as you conjectured. For $A_n$, there are a couple of exceptions: $A_7$ and $A_8$ have representations of degree $4$.