Cockney

Why are those born within the sound of Bow Bells, London called Cockneys?
Is it with reference to the Cocks Egg?

From Merriam Webster:

One entry found for cockney.

Main Entry: cock·ney
Pronunciation: 'käk-nE
Function: noun
Inflected Form(s): plural cockneys
Etymology: Middle English cokeney, literally, cocks' egg, from coken (genitive plural of cok cock) + ey egg, from Old English [AE]g
Date: 14th century
1 obsolete a : a spoiled child b : a squeamish woman
2 often capitalized a : a native of London and especially of the East End of London b : the dialect of London or of the East End of London
- cockney adjective
- cock·ney·fy /'käk-ni-"fI/ transitive verb
- cock·ney·ish /-nE-ish/ adjective
- cock·ney·ism /-"i-z&m/ noun

Can someone explain to a foreigner whether "Cockney" is an insult these days? Its history, set forth in the Oxford English Dictionary, shows that it was a derisive term when first applied to Londoners--but people don't seem to mind calling themselves Cockneys.