What's the transition from “metal” to “electron-degenerate matter” look like?

Question: Say that we had metallic water and compressed it until it qualitatively resembled the state of matter in a white dwarf: what would that transition look like?

Background (revised to be shorter)

In an answer, I claimed that, as water was pressurized, it'd freeze into a high-pressure ice, then metallize, then go through degenerate stages of matter, etc..

Commenters pointed out that I skipped mentioning a "white-dwarf" phase in which the water would be pressurized enough to be electron-degenerate, like the matter in a white dwarf is thought to be.

The trouble is that I'm not sure how to describe the difference between metallized matter and electron-degenerate matter. As I understand it, the electrons in both have been approximated to exist as a Fermi gas.

In the case of just-barely-metallized matter, presumably only electrons in the outermost electron shells are in the Fermi-gas state, whereas in full-fledged electron-degenerate matter, all electrons would be in the Fermi-gas state.

Questions (restated):

1. Is it accurate to say that the difference between these states is the portion of electrons in a Fermi-gas-like state?

2. What's the compression-driven phase transition from "metallized water" to "electron-degenerate water" look like?

Note: Obviously using "water" loosely here. If it's easier, an answer might refer to metallic hydrogen vs. electron-degenerate hydrogen instead (again noting that that may be a loose description, given fusion).

Background (longer version)

In response to a question about what'd happen if water was increasingly compressed, I'd written up an answer about how water would "freeze" into an ice, then metallize, then fuse, then go through degenerate matter phases, then probably end up as a black hole or something.

The comments had pointed out I skipped any mention of a "white-dwarf" phase, but I'm pretty fuzzy on how to see that transition; so, that's what this question is about.

As I understand it, water is predicted to metallize around ${10}^{12}\mathrm{Pa}$:

As Wikipedia describes metallization:

Metallic hydrogen is a kind of degenerate matter, a phase of hydrogen in which it behaves like an electrical conductor.

-"Metallic hydrogen", Wikipedia [formatting omitted]

According to electronic band structure, electrical conductors (metals) work the way that they do because their conduction band is partially filled,

$\hspace{3cm}$,

such that it requires extremely little energy to get an electron from a filled electronic state to an unfilled electronic state.

And, apparently, this can be well-effected to the point of super-conductivity through pressurized metallization, much like discussed for water above, though as recently demonstrated for $\text{H}_2\text{S}$ at $1.5{\cdot}{10}^{11}\mathrm{Pa}$:

In 2015, hydrogen sulfide (H2S) under extremely high pressure (around 150 gigapascals) was found to undergo superconducting transition near 203 K (-70 °C), the highest temperature superconductor known to date.

-"High-temperature superconductivity", Wikipedia [partial formatting; references omitted]

So, it seems that pressurized metallization can be used to justify the independent electron approximation, such as used in the free electron model, such that the electrons would seem to be a Fermi gas.

Then, more speculatively on my part, that'd seem to suggest that electron degeneracy pressure would apply here, much like it would in a white dwarf, since the electrons are describable as being in a Fermi gas either way. I suppose that there might be soft phase-transitions as a metallized material is further pressurized since, I assume, metallization tends to be a characteristic of the conduction band and not all of the molecular orbitals, such that there may be non-degenerate electrons in lower-pressure states.

Then, I guess, further pressurization would tend to increase the portion of electrons that're degenerate (as well as the quality of the degenerate approximation), and that the matter would tend to shrink (increase in density) like white dwarfs are predicted to do (since their density greatly increases with mass).

Wrapping that up

The main question's at the start of this post, but basically, is the above on-track? And either way, what's the conceptual difference between "metallized" and "electron-degenerate" matter, and what's the transition between them look like as pressure is increased?