What's the correct pronunciation of "Israel"?

The OED says that one says /ˈɪzreɪəl/.

But one sings /ˈɪzra(j)ɛl/ in the opening of Mendelssohn’s Elijah:

As God the Lord of Israel liveth, before whom I stand: there shall not be dew nor rain these years, but according to my word.

That’s the standard sung pronunciation; it (meaning /ˈɪzra(j)ɛl/) is perhaps what people are hearing as your “rye” thing. It is hardly barbaric.

The spoken pronunciations are what they are, and will be no matter what anybody says about them.

Your second link, however, is not about spoken pronunciation but about sung pronunciation in classical music; and that art form has its own conventions. Classical music insists on 'pure', 'Italian' vowels; it does not like diphthongs and glides, because these modify the 'pure' musical tone. This aversion is particularly acute among choral musicians, who want everybody to be singing the same vowel at the same time. Don't be misled by the spelling of the preferred pronunciation as <ay>; conscientious choristers know that this is merely an English approximation of /e:/.

In the Jewish community in the United States you hear Is-Ree-al and Is-RYE-el, the latter being closer to the Hebrew pronunciation of YIS-ra-el. The more imprtant distinction should be the "El" which is a form of a Name of the Divinity.