It is possible the videos are from the 0.95a release of the game. Phase ships and related skills underwent a fairly large change in behavior in the 0.95.1a release.
In 0.95.1a the following changes were made:
Phase cloak reduces top speed as hard flux goes up, down to 33% speed at 50% hard flux while phased
New hullmods:
Adaptive Phase Coils: increases hard flux level for minimum speed to 75%
Phase Anchor:
Reduces phase cloak activation flux cost to 0
Increases soft flux dissipation and weapon cooldown/charge regen by 2x while phase
One ship *per battle* can execute an "emergency dive" maneuver instead of being destroyed
Counts as retreated instead and loses an extra deployment's worth of CR
High ordnance point cost and can not be built in
In addition, Phase Mastery was removed, and it's elite version +100% speed bonus became a +50% speed bonus in the Phase Coil Tuning Tier 3 Technology fleet wide skill.
So phase ships have a lower maximum speed these days assuming you took all relevant skills before and after, and slow down the more flux they build up, to the point where you're only moving at 1/3 of your speed (or effectively normal speed when you consider the x3 speed up).
So far a Doom, to maximize speed you can:
Take Elite Helmsmanship (+10% speed, +5 base speed)
Take Phase Coil Tuning for +50% speed, and make sure total number of combat phase ships is at 40 DP or less (so basically just a Doom in your fleet)
Get 100% CR for +10% speed, Combat Endurance + Crew Training for example
Take Coordinated Maneuvers and deploy a couple frigates with officers (up to +20% speed)
Add the Unstable Injector hullmod (+15 base speed)
Add the Adaptive Phase Coils hullmod which reduces the speed penalty due to hard flux somewhat.
I don't recommend putting safety overrides on a Doom, although you can do so for a huge speed boost since it gets the 0-flux +50 speed bonus even when phased, along with a base +20 speed, but you will run out of peak performance time very quickly.
Anwyays that is all you can do to try and approach the speed of the old Doom with skills, but will still fall short of that old speed, especially once you've hit something like 30-40% hard flux and are taking a -33% speed penalty even with Adaptive Phase Coils.
However, Phase Anchor opens up another possibility, which is a Doom which just multiplies it's DPS by not quite a factor of 6, through the 3x speed effect for being in phase, and the 2x weapon cooldown/flux dissipation effect. This lets you use weapons with long cooldowns (or a charge mechanic) but high burst (Antimatter blasters, Needlers, Phase Lances, Ion Pulsar, etc) to good effect. Essentially you can think of it being able to fire 6 times (or maybe more like 4 or 5 in actual practice) in the space of time a normal ship could fire them once. It's not going to be as fast, but it can in principle roll up, fire, cloak and cooldown weapons, fire, cloak and cooldown weapons, and so on. Elite Field Modulation is very valuable in this case, as it reduced the phase cooldown by 50%.
