The website is an informative portal thats been around for 20 odd years, it is no way "formal"... the formal language is in the IRBs regulations, laws and by-laws, which have been around for well over 100 years.. and they define the game as "Rugby Football", THAT is its FORMAL
Those definitions include:
Agent means an agent or adviser acting on behalf of a Person, Union, Rugby Body or Club in relation to that Persons, Unions, Rugby Bodys or Clubs activity in the Game.
Game means rugby football played in accordance with the Laws of the Game.
IRB means the International Rugby Board, being the association of Unions or Associations in membership of the Board in accordance with the Bye-Laws.
International Tour means a Match or Series of Matches in which a team representing a Union at any level participates against a National
Representative Team and/or other Rugby Bodies or Clubs in the
jurisdiction of another Union.
IRB Sevens Tournament means the Rugby World Cup Sevens (Finals and Qualifying Matches), the World Sevens Series and such other Sevens Tournaments that the IRB shall from time to time designate.
Rugby Body means a provincial Union, district, state, group of Clubs or similar organisation affiliated directly or indirectly to a Union.
Union(s) means every national rugby Union for the time being in membership of the Board.
It seems that any time the games name is defined in the IRBs regs its Rugby or Rugby Football... the only time Rugby and Union are together its talking about a governing body. Rugby IS the games name... it is fully entitled to use it in that sense.
FFS that is because they are colloqualising in the regs, they clearly state that they governing and law making body of the game of Rugby Union, so the game must be called Rugby Union not just Rugby