The distinction between a completely metrizable space and a complete metric space lies in the words there exists at least one metric in the definition of completely metrizable space, which is not the same as there is given a metric (the latter would yield the definition of complete metric space).
Once we make the choice of the metric on a completely metrizable space (out of all the complete metrics compatible with the topology), we get a complete metric space.
[5] When talking about spaces with more structure than just topology, like topological groups, the natural meaning of the words “completely metrizable” would arguably be the existence of a complete metric that is also compatible with that extra structure, in addition to inducing its topology.
For abelian topological groups and topological vector spaces, “compatible with the extra structure” might mean that the metric is invariant under translations.
Indeed, a topological vector space is called complete iff its uniformity (induced by its topology and addition operation) is complete; the uniformity induced by a translation-invariant metric that induces the topology coincides with the original uniformity.