For non-Hermitian quantum systems, metric operators provide a way to define the corresponding inner-product structures. In the entirely real-spectrum regime, a positive-definite metric $\eta$ can be introduced such that the Hamiltonian is $\eta$-self-adjoint with respect to the corresponding inner product. In the complex-spectrum regime, different metric constructions can be characterized by their associated norm properties. For finite-dimensional non-Hermitian systems with a complete energy eigenbasis, we compare three representative metrics, namely the cross-inner-product metric ${{\rm Y}}_1$, the dynamical metric ${{\rm Y}}_2(t)$, and the pseudo-Hermitian metric ${{\rm Y}}_3$, and analyze their relationships and differences. For time-independent Hamiltonians, the ${{\rm Y}}_1$ norm obeys a balance equation governed by the imaginary parts of the eigenvalues, with its long-time behavior dominated by the largest imaginary part among contributions with nonzero initial weights. In contrast, ${{\rm Y}}_2(t)$ preserves the norm through its time dependence, while ${{\rm Y}}_3$ remains norm-preserving but becomes indefinite in the complex-spectrum regime. For an entirely real spectrum, all three metrics coincide under the same normalization convention when ${{\rm Y}}_2(t_0)={{\rm Y}}_1$. At exceptional points, the energy eigenbasis is incomplete and ${{\rm Y}}_1$ is singular and positive semidefinite, whereas a positive-definite ${{\rm Y}}_2(t)$ can be obtained from another complete initial basis, and a Hermitian invertible ${{\rm Y}}_3$ can still be obtained from its defining equation. For time-dependent Hamiltonians, the ${{\rm Y}}_1(t)$ norm equation acquires a geometric contribution from the evolving biorthogonal basis, while imposing dynamical compatibility on ${{\rm Y}}_3(t)$ makes it coincide with ${{\rm Y}}_2(t)$ for the same initial metric. The corresponding exceptional-point properties are also discussed. A concrete model illustrates these distinctions.