The Christmas Island in the Pacific is the one where the UK conducted the tests, and afterwards the USA did a few tests there also as an agreement between the UK and USA, along the lines of --if the UK has access to the Nevada Test Site for underground detonations, then they (the USA) can use Christmas Island....-- sort of thing.
It is part of the Gilbert/Ellice islands of which Cook discovered in 1777 and thereafter annexed as part of Britain.
The name is now changed to 'Kiritimati', part of the Kiribati atoll, the largest in the pacific.
The Christmas island in the Indian Ocean belongs to Australia, and is where they like to send all their illegal immigrants.....atleast the ones that were onboard the freighter a few years ago anyway.