The issue of the names Hungarians and Magyars rests in understanding of the society before the ideology of Nationalism arose in the Kingdom of Hungary /before the end of the WW1/.
Magyars = an ethnic group of people, nowadays a modern nation. In Magyar language there is no term Hungarian, they took it over but only in English, Latin, German and other languages.
Hungarians = at first there was so-called Natio Hungarica, which wasn't any ethnic description, but it was a term for the aristocracy regardless of nationality/language. All the others were villeins also regardless of nationality/language. Then it started to be considered as a citizenship – again rather than nationality. So a man was ethnically Magyar/Slovak/Croatian/Ruthenian etc., but he was citizen of the Kingdom of Hungary, so he was the Hungarian (just like Franz Liszt – ethnically Slovak/Austrian, but Hungarian by citizenship – and he absolutely didn't speak Magyar). Then we come to the time of Romanticism and Nationalism, when Magyars started to identify with the term Hungarians (probably because of its primordial noble meaning, although most of Magyars were villeins just like the others).