The country I live in is "The United Kingdom of Great Britain and Northern Ireland" (a nice long winded title for several quite small pieces of land), which includes the mainland, Northern Ireland all the Scottish islands and the Isle of Wight. It does not, however, include the Isle of Man or the Channel Islands (or, for that matter, the Falklands, Bermuda and a few other odd bits of Empire left lying around), these are British Crown Dependent Territories. Unfortunately I have not been able to find a good definition of a CDT, but suffice to say that they have a share of being "British" whilst retaining some independence. They are not (as far as I know) capable of becoming members of the UN and as such are not "true" countries in their own right.
"Great Britain" is a simple definition of the mainland (England, Scotland and Wales) and tends to be a bit misleading since people use it interchangeably with "United Kingdom".
"The British Isles" is a definition of all the islands around North West Europe that are or were at some point directly controlled by the British Crown, including Great Britain, All Ireland, The Isle of Man and The Channel Islands.
"England" (& "Scotland" & "Wales") are regions defined for historical interest only and are no more separate countries (or any more culturally diverse) than (say) Yorkshire. Although with devolution, it might be argued that Wales and Scotland could theoretically achieve the status of CDT at some point in the future (and who could say whether we'd be better off without them anyway <g>).
The WCF has its own agenda for defining countries, including (and probably not limited to) the fact that to attempt to enter large diverse "games" competitions (Olympics etc.) the more countries that you can claim play for you the better the chance of being allowed in. (This policy has not however worked, to my knowledge). As a consequence we have a raft of regions which are not true countries but which people are allowed to represent.
This again is historically complex since it used to be the entire British Isles, but over time got eroded (the Irish CA were formed a long time ago, The Scottish and Welsh CAs more recently) and now covers a very bizarre area which overlaps with other CA's territories in some cases. From the CA's website: "The Croquet Association (CA) is the national governing body for the sport of croquet in England, Wales, Northern Ireland, the Channel Islands and the Isle of Man and is responsible for promoting and regulating the game in these areas."
In conclusion, under a strict definition of "Great Britain", Matt is not entitled to play, under a loose definition of "Great Britain" (as a synonym for "United Kingdom") he may be able to play, under the WCF's rules he is not entitled to play and under the CA's rules he is allowed to play.
I hope this makes everything clear.