The Fellowship in Valinor

[Arothir]

The "immortals" who were permitted to leave Middle-earth and seek Aman-the undying lands of Valinor and Eressea...

As for Frodo or other mortals, they could only dwell in Aman for a limited time...

This seems to suggest that the Fellowship actually lived in Aman, the mainland. However, according to the first quote from Letters, Aman is Tol Eressea and Valinor. I thought Aman was just the mainland.

...Frodo may as a special grace be granted a purgatorial sojourn in Eressea, in sight of Aman.

This is even more perplexing. Here, Frodo clearly stays at the Lonely Isle, which is not part of Aman. Is there something I'm not getting?