The Greek name for Greece is Hellas, which is composite from Helios and Laas, which would translate as 'Sunland'.

At the ancient times, Greeks would call 'Sunland' the region around mountain Parnassus, which was considered to be the mountain of god Apollo, god of the Sun, and lies in about the middle of modern Greece. (See how the land of the Sun is related with the God of the Sun) ?

The Greeks also considered themselves to be descendants of Deucalion and Pyrra, who according to the myth, after a worldwide flood, their ark landed on mount Parnassus. So the myth says that these two stayed there and had children, and thus Hellenes (how Greeks called each other) also means "Sunlandfolk", as (according to the myth) they were children of this couple, born at Parnassus.

By the time of Alexander the Great, Greece could be considered in a way his empire. Greece as in that it was ruled by Greeks.

After Alexander and before the Roman Empire, there were 3 Greek kingdoms, one that consisted of modern Greece and parts of Anatolia, another that included Egypt and the Middle East (Syria, Israel, Lebanon,etc..) and one from Persia to India.

During the Roman times, Greeks identified as Greek Romans, and during the Byzantine times, Greeks just called themselves Romans.

When the Ottomans invaded, they called Greeks 'Rum', from the word Roman. The Byzantine Empire had already been weakened though by a number of other invaders like the Crusaders.

During the Ottoman era, the area at the south of Wallachia, was called 'Rumilia', meaning 'Roman Land'.

Where did you read that Anatolia was Greek? I'm honestly interested. Some coastal regions in the far west or southwest of the peninsula had large Greek minorities and may have been considered part of a Greek nation in the 1800s and on, but not Anatolia at large. Of course, I could always be wrong or there could still be debate on the topic.