Your comparison of Bahamian island Eleuthera with Caribbean island Anguilla is really very interesting. I'm saying this because I was never in Anguilla but was wondering how it's different. Web sites don't always show the true nature of an island!
Here on Eleuthera I have met Winter residents who had moved from Anguilla to Eleuthera because Anguilla seemed too developed to them.
On one hand Eleuthera is not as touristic as Anguilla and much closer to Florida - only an hour by air - and on the other hand Anguilla offers more high-end resorts. No wonder that international celebritiy prefers Anguilla.
But famous musicians like Mariah Carey and Lenny Kravitz chose Eleuthera. They seem to like the semi-developed nature of this Bahamian island.