Maybe it would help to understand the terms being used. One of the problems is that people are introduced to Buddhism without it being fully translated. I am working on a translation project that should help to remedy this by having all such terms translated from the start. But maybe the following will help:
Enlightenment comes from the word bodhi
, which actually means awakening
. It comes from the root budh
is the past participle of budh
, which thus in English means awakened
. If it is being used to describe someone/something, it is translated as Awakened One
means extinguishing, or cessation
, just like you would extinguish a flame, or make it cease to flame.
The Awakened One
is therefore one who has achieved awakening
. Awakening means seeing the four noble truths directly: Suffering, the cause of suffering, the cessation of suffering, and the path to the cessation of suffering.
One who has attained cessation
is one who causes both the skandhas (aggregates) and kleśas (defilements) to cease
. Thus ending the causes for continued rebirth.
One attains cessation by achieving awakening: seeing the four noble truths directly, and by seeing the causes of suffering that knowledge allows them to cease producing the further causation of suffering. They thus cause the defilements to cease. Thus that person is called a Foe-Destroyer (Arhat) since they have destroyed their foes, the defilements. Still having the aggregates and karma to be worked out, one has attained Cessation with Remainder.
When both body and mind have ceased to continue, and the aggregates and karma no longer need to be worked out, one attains Cessation without Remainder
Let's see if I can use this framework to answer your questions:
I can't figure out how that ties in with Enlightenment and Buddhahood. Is Buddhahood achieved once you're completely liberated from samsara/reach Nirvana?
No, being an Awakened One is not the same as achieving cessation. One who achieves awakening when there is currently no Buddhist dispensation in the world is the Awakened One of that era. After that, all who achieve awakening without vowing in the future to themselves become an Awakened One will become a Foe-Destroyer. You attain awakening prior to being completely liberated from Samsara. If you make the vow to become an Awakened One, you will put off full cessation until after that time, and engage upon the path of the Awakening Hero (Bodhisattva) which may take many eons. However, you may very well be awakened during that time, and may achieve cessation (at what is called the Seventh Level of the Awakening Hero's path), but due to one's vow, one doesn't enter Final Cessation (parinirvāṇa) until in the future one attains the state of an Awakened One. The full awakening of an Awakened One is called Unexcelled Perfect and Complete Awakening (anuttarā-saṃyak-saṃbodhi), which is more extensive than that of a Foe Destroyer.
The state of a Foe-Destroyer is much easier to attain, and can be done in one life, whereas the state of an Awakened One takes many lives to work out. This is why it is important for some to make the vow to become an Awakened One in the future when there is no dispensation in the world, or in worlds where there is no current dispensation.
If so, is Enlightenment a preliminary requisite to Buddhahood?
Yes, to be an Awakened One, you need to achieve awakening. This is why I believe it's so important to have these terms translated - it becomes apparent when it's in English.
Or is Enlightenment mutually inclusive with Nirvana?
Awakening causes cessation, but not if you vowed to delay cessation until you attain the state of an Awakened One.
If that's true, then what's the functional difference between a Buddha and a Boddhisattva if once you become Enlightened you reach Nirvana even though Boddhisattvas renounce Nirvana?
An Awakened One has already achieved that state, and an Awakening
Hero (Bodhisattva) is aiming for that state. Awakening Heroes only renounce individual cessation, they aim for the state of an Awakened One out of compassion for the suffering of all sentient beings - they want to teach others so they can relieve them from suffering. When they achieve the state of an Awakened One, then they can achieve Final Cessation, just like the Awakened One in this era, Shakyamuni.
There is more to it, but it's best to take it one step at a time.