Some background to taxonomy of cacti
Over the last century there have been a number of revisions of the taxonomy (the naming and classifying) of Cacti.
The most recent revision of cactus taxonomy was published in the New Cactus Lexicon by David Hunt in 2006. http://www.newcactuslexicon.org
Why you may like to know the official name
it is useful to know what is the official name of a cactus or succulent when you are looking at plant and seed lists, or for information on their requirements.
It is especially handy to know whether the name you are using is treated as a synonym and if so, what is the official name.
ITIS webpage - A handy web based reference for plant taxonomy
One of the easiest ways to check for the official name of a cactus is to use the Integrated Taxonomic Information System (ITIS for short). http://www.itis.gov/
This first page is the launching point for any search you do, so putting cactacea in the scientific name box will give you a summary reporting page of the accepted genus names for cacti.
Alternatively you can enter the name of a plant and the result will tell you if the name is accepted or is a synonym, and what the correct name is.
Happy name hunting!