the IARU list portray the heritage of a hundred years. It is not so strange then that this may make some of them hard to understand and even a bit old-fashioned.
Starting with the ARRL - American Radio Relay League or NRRL (Norwegian ...) then this is about a network of stations relaying messages in a country with large distances. This is a bit 1920's to me. Looking at the ARRL web page it looks as if the the ARRL agrees and really would like to modernize the name to the National Association for Amateur Radio. The Portuguese may already have modernized it a bit by the use of network instead of relay: "Rede dos Emissores Portugueses" (Network of Portuguese Transmitters), or perhaps it is just because this is the same word as "relay" in Portuguese?
Most societies have "radio amateur" in the name like the German and the French: Deutscher Amateur Radio Club, Union Francaise des Radioamateurs. My knowledge of Spanish is not so good but this one sounds better to me: Union de Radioaficionados Espanoles as "radioaficionados" gives me the impression of "radio fans", but perhaps it is just another word for "radio amateurs"?
Just like "radio relay" is a bit old-fashioned, the Swiss also have an old name in Union Schweizerischer Kurzwellen-Amateure. It must have been coined before VHF and UHF as Kurzwelle means Shortwave.
A name which sounds more contemporary despite its age is the Radio Society of Great Britain. Many others have similar names, one example is Amateur Radio Society of India.
The word "club" is used by many and seems to emphasize the hobby aspects, e.g. Radio Club Argentino and Český radioklub (Czech Radio Club).
Contesting is in many ways like a sport, at least in the same sense as chess is called a sport today. Examples that use this word are Chinese Radio Sports Association, Belarussian Federation of Radioamateurs and Radiosportsmen, and Mongolian Radio Sports Federation.
The there's those who value the experimental aspects, like Vereniging voor Experimenteel Radio Onderzoek in Nederland (Society for Experimental Radio Research), Experimenterende Danske Radioamatører (literary Experimenting Danish Radio Amateurs) and Federacion Mexicana de Radio Experimentadores.
A similar technical emphasis is in the word "transmitter": Irish Radio Transmitters Society, Foreningen Sveriges Sändareamatörer (literary Society of Swedish Transmitter Amateurs) and Österreichischer Versuchssenderverband (literary Austrian Experimental Transmitter Society).
The protection from the royal family exalts the society, but it is something I could only find in Belgium: Royal Union Royale Belge des Amateurs-Emetteurs / Koninklijke Unie van de Belgische Zendamateurs / Königliche Union der Belgischen Funkamateure.
Finally, these are the most serious and ambitious names I could find: Egyptian Radio Amateurs Society for Development, Syrian Scientific Technical Amateur Radio Society, and above all the Wireless Institute of Australia.
Which words should be used in the name of a radio society today? It's a balance between reflecting a hundred years of history and communicating with contemporary people.
Personally I don't like to use the word "amateur". It comes originally from the same root as "amour" and "amore" and thus means someone who loves and is passionate about something. But today it gives the impression of being unprofessional. A contemporary name should also emphasize the experimental side in my view.