The Anglosphere is the English speaking world. It could further be described as an area with a shared history and cultural customs. I find the second part somewhat problematic though, since that assumes that to be an English speaker, someone must be connected to some essential "Englishness", or it just begs the question by defining the culture of English speakers as the culture of people who speak the English language, and defines the English language as that spoken by people with English cultural ties. Wishing to avoid the cultural issues, I will define the Anglosphere in more practical terms. I do however, agree that the Anglosphere has concentric levels, with some countries in the core and some countries on the periphery.
The most core Anglosphere countries are usually considered to be The United Kingdom, the United States of America, Canada, Australia and New Zealand. I base this on the fact that, as an English as a Second Language teacher, most schools looking for a native speaker will specify a passport holder of those countries. There are a number of other countries where English is spoken almost as universally, but are not considered to be quite as culturally integrated as the main five: South Africa, Ireland and Jamaica. Notice that this is not a matter of clarity of pronunciation: the average Jamaican is probably easier for a "standard" English speaker to follow than someone from Scotland, Alabama or Australia, but due to prejudice, many people consider them to speak a "secondary" English. In the next ring out are countries where English is widely used as a secondary or official language, but where people have another, stronger language or cultural identity. This could include countries like The Philippines, Singapore, India or Israel. Then there are a few countries where English is learned as a second language, but to the degree that it is spoken with native precision: in most of Scandinavia and The Netherlands for example. And in our furthest circle is countries where English is mostly spoken as a language of academics and business, but is not commonly spoken: Russia and Japan, for example.
While the exact placement of a country in these categories might be a matter for discussion, but the general nature of how English is used in different regions is probably clear. It also raises the issue of how we look at language and nationality: If only 10% of the population of India was raised as English speakers, it still means it has twice as many native English speakers as the United Kingdom. At what point does the English spoken by people who "aren't English speakers" start to outweigh the English spoken by those who are?
And, as a personal note, this is a list of (some) of the countries my fellow English teachers in Chile have come from: The United States, The United Kingdom, Canada, New Zealand, Australia, Ireland, South Africa, Russia, Poland, Hungary, The Philippines, Malaysia, India, Brazil, Venezuela and Chile.