= Solution
For the spatial supremum-norm convention, a normalized <Hamiltonian function> gives the path length
$$
\ell_\infty(H)=\int_0^1\|H_t\|_\infty\,dt,\qquad \|H_t\|_\infty=\sup_{x\in M}|H_t(x)|.
$$
On an open manifold use compactly supported Hamiltonians; on a closed manifold use zero-mean Hamiltonians, which fixes their additive time-dependent constants. The <Hofer metric from the spatial supremum norm> is
$$
\boxed{\rho_\infty(\phi,\psi)=\inf_{\phi_H^1=\phi^{-1}\psi}\ell_\infty(H)}.
$$
Equivalently one minimizes this length over Hamiltonian paths joining the two maps. Inversion and conjugation preserve the generator's supremum norm, so the distance is symmetric and bi-invariant; concatenation gives the triangle inequality. Its nondegeneracy is the geometric assertion proved in Question 3, not a consequence of these formal properties alone.
Another common convention for <Hofer's metric> uses the oscillation length $\ell_{\mathrm{osc}}(H)=\int_0^1(\max H_t-\min H_t)\,dt$, giving a distance $d_H$. Normalized or compactly supported Hamiltonians have values straddling zero, so
$$
\|H_t\|_\infty\leq\operatorname{osc}H_t\leq2\|H_t\|_\infty,\qquad \rho_\infty\leq d_H\leq2\rho_\infty.
$$
Thus these conventions give equivalent distances, and agree on whether an energy vanishes. We keep the same chosen convention in every energy estimate.
The <Hamiltonian displacement energy> of a subset $A$ is
$$
\boxed{e(A)=\inf\{\rho_\infty(\mathrm{id},\phi):\phi\in\operatorname{Ham}_c(M,\omega),\ \phi(A)\cap A=\varnothing\}}.
$$
The infimum is $+\infty$ if there is no displacing map. In the oscillation convention replace $\rho_\infty$ by $d_H$.
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