  We take a careful look at two approaches to deriving stability criteria for
ideal MHD equilibria. One is based on a tedious analysis of the linearized
equations of motion, while the other examines the second variation of the MHD
Hamiltonian computed with proper variational constraints. For equilibria
without flow, the two approaches are known to be fully consistent. However, for
equilibria with flow, the stability criterion obtained from the constrained
variation approach was claimed to be stronger than that derived using the
linearized equations of motion. We show this claim is incorrect by deriving and
comparing both criteria within the same framework. It turns out that the
criterion obtained from the constrained variation approach has stricter
requirements on the initial perturbations than the other. Such requirements
naturally emerge in our new treatment of the constrained variation approach
using the Euler-Poincar\'e structure of ideal MHD, which is more direct and
simple than the previous derivation from the Poisson perspective.
