After having a look at Silverman's "Arithmetic of Elliptic Curves", I mostly understand the notion of a supersingular Elliptic Curve and its characterizations. However, some subtleties still confuse me. Could someone point out the error in the following, obviously wrong argument?
Let $E$ be a supersingular elliptic curve defined over a $\mathbb{F}_{p^2}$, and assume further that $j(E) \neq 0, 1728$. By one of the characterizations of supersingularity, we see that the multiplication-by-$p$ map is purely inseparable.
By basic algebraic geometry (or e.g. theorem II.2.12 in Silverman's AoEC), it follows that $[p]$ factors as $$ E \ \overset{\pi^2}{\to} \ E^{(p^2)} = E \ \overset{\lambda}{\to} \ E $$ for the $p$-th power Frobenius morphism $\pi$ and a separable isogeny $\lambda$. However, since $[p]$ has separability degree 1, $\lambda$ is already an automorphism. Since $j(E) \neq 0, 1728$ we see already that $\lambda = \pm [1]$. So $\pi^2 = \pm [p]$, and in particular, the trace of the Frobenius endomorphism is $0$, as $\pi^2 \in \mathbb{Z}$.
Since all supersingular j-invariants are defined over $\mathbb{F}_{p^2}$ and the endomorphism ring of $E$ is preserved under isomorphism, this assumption should not make a difference.
However, I am sure that it is not true that the trace of the Frobenius endomorphism is always $0$ in case of a supersingular curve (at least, Silverman and Wikipedia say $E$ supersingular iff the trace is $\equiv 0 \ (\mathrm{mod} \ p)$).