Reincarnation according to Guenon

Can someone explain Guenon and Coomaraswamy's explanation of the doctrine of reincarnation to me? It seems I've fallen for the western meme version, but I can't wrap my head around their explanations.
More specifically: In the western meme version, it seems clear to me why one should wish to attain liberation. But if that version of reincarnation is wrong, I don't understand.

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With regard to the reincarnation question that's a point many people get confused about. What Guenon meant was that Vedanta teaches transmigration but not reincarnation, which he considered to be a western misunderstanding.

Reincarnation = Rene after death is reconstituted in another body with the same personality, mental attributes, etc in the next life; the individual essence of "Rene" continues

Transmigration = the subtle body (which has no awareness or consiousness itself but which is given activity and seeming awareness by the light of the Atma/Self) continues onto the next life, although everything that formed "Rene" as a unique identity is forever lost, the subtle body being no more Rene than it was the innumerable people that it passed through previously

This point is illustrated in this passage by Coomaraswamy from the Chapter 'The One and Only Transmigrant" in his book 'Perception of the Vedas':

Sankaracharya's dictum, ‘Verily, there is no other transmigrant but the Lord’ (satyam, nesvarad anyah samsari, BrSBh, 1.1.5) startling as it may appear to be at first sight, for it denies the reincarnation of individual essences, is amply supported by the older, and even the oldest texts, and is by no means an exclusively Indian doctrine. For it is not an individual soul that Plato means when he says: ‘The soul of man is immortal, and at one time comes to an end, which is called dying away, and at another is born again, but never perishes .. . and having been born many times has acquired the knowledge of all and everything’; or that Plotinus means when he says: There is really nothing strange in that reduction (of all selves) to One; though it may be asked, How can there be only One, the same in many, entering into all, but never itself divided up; or by Hermes who says that ‘He who does all these things is One, and speaks of Him as ‘bodiless and having many bodies, or rather present in all bodies’. The ‘Lord’ of whom Sarikaracarya speaks is, of course, the Supreme and Solar Self, Atman, Brahma, Indra, of all beings Overlord, of all beings King’, whose omniformity is timeless and whose omnipresence enables us to understand that He must be omniscient (sarvanubhuh, BUt II.5. 19, cf. IV.4.22 and SA, XIII); Death, the Person in the Sun, Indra and Breath of Life, ‘One as he is Person there, and many as he is in his children here’, and at whose departure ‘we’ die (SB, X.5.2.13, 16); the Solar Self of all that is in motion or at rest (RV, 1.115.1); our Immortal Self and Inner Controller ‘other than whom there is no seer, hearer, thinker or knower' (BU, III.7.23, III.8.11); the solar Indra of whom it is said that whoever speaks, hears, thinks, etc., does so by his ray (JUB, 1.28. 29); Brahma, of whom it is said that our powers ‘are merely the names of his acts’ (BU, 1.4.7, cf. 1.5.21); the Self, from whom all action stems (BU, 1.6.3; BG, 111.15); the Self that knows everything (MU, VL7). Whether as Surya, Savitr, Atman, Brahma, Agni, Prajapati, Indra, Vayu or madhyama Prana-yadrgeva dadrse tadrg ucyate (RV, V.44.6)6—this Lord, from within the heart here, is our mover, driver and actuator and whole source of the evanescent consciousness u that begins with our birth and ends with our death (MU, II.6d, III. 3).12 We do nothing of ourselves and are merely his vehicles, and instruments (as for Philo, passim). This ‘higher’ (para) Brahma is that ‘One, the Great Self, who takes up his stand in womb after womb .. .as the omniform Lord of the Breaths'.

Thank you. I am trying to wrap my head around that quote.
What I really struggle with is liberation: Can you perhaps explain that to me? What exactly is liberated?
>Transmigration = the subtle body (which has no awareness or consiousness itself but which is given activity and seeming awareness by the light of the Atma/Self) continues onto the next life
"This" subtle body? Or rather: "This" subtle body stops being (seeing itself as?) a distinct thing? Whereas other subtle bodies "remain"?

this is difficult stuff, but i'll try to help

>What exactly is liberated?

nothing is actually liberated, however the misidentification with the body and the subtle body are dropped away, and all that is left is identification with god

the subtle body is what most people consider to be the 'soul', as it contains the knowledge and impressions of the past lives and of you right now. but since even that is in a constant state of flux, the subtle body isn't you either. anything in a state of flux/change is maya. so when you go far enough to even remove the subtle body, you just have god as the only transmigrant

mahtalcar.rusff.ru/viewtopic.php?id=7294

Thank you. I see that I still have a lot to read and to wrap my head around. I'm not sure if this answers the question I had, largely because I'm not sure I even understand the question I had.
Maybe: What difference does it make (past death in this realm) if "I" attain liberation or not?

Cпacибo! Я бyдy cтapaтьcя, нo мoй ypoвeнь pyccкoгo языкa нe oчeнь выcoкий.

Just translate that page, bro.

>What difference does it make (past death in this realm) if "I" attain liberation or not?
Because then you just exist as pure bliss forever instead of subjecting yourself to the illusion of embodiment and all the problems and suffering that comes with this.

How does Guenon reconcile this with his Islamic beliefs?