Both points are (mostly) correct, but there are more reasons.
The Summoning of the Elves
I would say this one is more correct. The Valar had extended an open invitation to the Elves, which they did back in the First Age (emphasis mine):
Then again the Valar were gathered in council, and they were divided in debate. For some, and of those Ulmo was the chief, held that the Quendi should be left free to walk as they would in Middle-earth, and with their gifts of skill to order all the lands and heal their hurts. But the most part feared for the Quendi in the dangerous world amid the deceits of the starlit dusk; and they were filled moreover with the love of the beauty of the Elves and desired their fellowship. At the last, therefore, the Valar summoned the Quendi to Valinor, there to be gathered at the knees of the Powers in the light of the Trees for ever; and Mandos broke his silence, saying: 'So it is doomed.' From this summons came many woes that afterwards befell.
The Silmarillion III Quenta Silmarillion Chapter 3: "Of the Coming of the Elves and the Captivity of Melkor"
At the end of the First Age the Elves were invited to return1:
Thus an end was made of the power of Angband in the North, and' the evil realm was brought to naught; and out of the deep prisons a multitude of slaves came forth beyond all hope into the light of day, and they looked upon a world that was changed. For so great was the fury of those adversaries that the northern regions of the western world were rent asunder, and the sea roared in through many chasms, and there was confusion and great noise; and rivers perished or found new paths, and the valleys were upheaved and the hills trod down; and Sirion was no more.
Then Eönwë as herald of the Elder King summoned the Elves of Beleriand to depart from Middle-earth.
The Silmarillion III Quenta Silmarillion Chapter 24: "Of the Voyage of Eärendil and the War of Wrath"
Letter 131 suggests that this "invitation" wasn't quite as cordial as it sounds:
We learn that the Exiled Elves were, if not commanded, at least sternly counselled to return into the West, and there be at peace. They were not to dwell permanently in Valinor again, but in the Lonely Isle of Eresseëa within sight of the Blessed Realm.
The Letters of J.R.R. Tolkien 131: To Milton Waldman. 1951.
Of course, we know that some chose not to return, and they lead us to the second point.
The Fading of the Elves
It's not wholly inaccurate to say that the "Fading" of the Elves was the result of their bodies being consumed by their souls. The theology of Middle-earth has the notions of fëa and hröa, respectively spirit and body. Keep those in mind as you read the following excerpt:
As ages passed the dominance of their fëar ever increased, 'consuming' their bodies (as has been noted). The end of this process is their 'fading', as Men have called it; for the body becomes at last, as it were, a mere memory held by the fëa; and that end has already been achieved in many regions of Middle-earth, so that the Elves are indeed deathless and may not be destroyed or changed.
History of Middle-earth X Morgoth's Ring Part 2 "The Second Phase" Chapter 3: "Laws and Customs Among the Eldar" Of Death and the Severance of Fëa and Hröndo [> Hröa]2
However, I suspect this was a minority case. The only way for this to occur is if an Elf manages to not die in Middle-earth (by which I mean, avoid the destruction of their hröa), which is not an easy task.
The Death of the Elves
Incidentally, the destruction of the hröa is yet another way many Elves could have gotten back to Aman:
Those fëar, therefore, that in the marring of Arda suffered unnaturally a divorce from their [hröar] remained still in Arda and in Time. But in this state they were open to direct instruction and command of the Valar. As soon as they were disbodied they were summoned to leave the places of their life and death and go to the 'Halls of Waiting': Mandos, in the realm of the Valar.
If they obeyed this summons different opportunities lay before them. The length of time that they dwelt in Waiting was partly at the will of Námo the Judge, lord of Mandos, partly at their will. The happiest fortune, they deemed, was after the Waiting to be re-born, for so the evil and grief that they suffered in the curtailment of their natural course might be redressed.
History of Middle-earth X Morgoth's Ring Part 2 "The Second Phase" Chapter 3: "Laws and Customs Among the Eldar" Of Death and the Severance of Fëa and Hröndo [> Hröa]
This fate got at least some of the Úmanyar (in some earlier drafts, the Alamanyar), the Elves (and descendants of the Elves) who began the trip to Aman in the First Age but never made it, to Aman3:
Concerning the fate of other elves, especially of the Dark-elves who refused the summons to Aman, the Eldar know little. The Re-born report that in Mandos there are many elves, and among them many of the Alamanyar
History of Middle-earth X Morgoth's Ring Part 2 "The Second Phase" Chapter 3: "Laws and Customs Among the Eldar" Of Re-birth and Other Dooms of Those That Go To Mandos
The Weariness of the Elves
Being an immortal creature in mortal lands gets boring after a few hundred years, so in the Second and Third Ages some Elves, who didn't return to Aman at the end of the First Age, just got tired of Middle-earth and sail West:
There Amroth and Nimrodel held a long debate; and at last they plighted their troth, "To this I will be true," she said, "and we shall be wedded when you bring me to a land of peace." Amroth vowed that for her sake he would leave his people, even in their time of need, and with her seek for such a land. "But there is none now in Middle-earth," he said, "and will not be for the Elven-folk ever again. We must seek for a passage over the Great Sea to the ancient West." Then he told her of the haven in the south, where many of his own people had come long ago. "They are now diminished, for most have set sail into he West; but the remnant of them still build ships and offer passage to any of their kin that come to them, weary of Middle-earth. It is said that the grace that the Valar gave to us to pass over the Sea is granted also now to any of those who made the Great Journey, even if they did not come in ages past to the shores and have not yet beheld the Blessed Land."
Unfinished Tales Part 2: The Second Age Chapter IV: "The History of Galadriel and Celeborn" Amroth and Nimrodel
1 This re-invitation is fairly important, because the Elves who left Aman in the First Age (the Noldor) did so under unpleasant circumstances and were barred from returning. This is part of the Doom of Mandos (or the Prophecy of the North):
'Tears unnumbered ye shall shed; and the Valar will fence Valinor against you, and shut you out, so that not even the echo of your lamentation shall pass over the mountains. On the House of Fëanor the wrath of the Valar lieth from the West unto the uttermost East, and upon all that will follow them it shall be laid also.
The Silmarillion III Quenta Silmarillion Chapter 9: "Of the Flight of the Noldor"
2 This notation means that Tolkien's initial choice for the word, hröndo, was later changed to hröo
3 If you're not familiar with the tribes of the Elves, Legolas and his people are counted among the Úmanyar.