Магнитофон is actually a name of the device that plays audio records that are written to a magnetic tape. It could be a cassette player, or a reel-to-reel tape recorder. If you want to disambiguate between these two kinds, you can call them кассетный магнитофон or катушечный (бобинный) магнитофон respectively (I guess this term become popular because of WWII german Magnetophon device).
Магнитофон is single-source device - it can play or record audio. Nowadays most devices can play audio from many sources - from CD's, mp3 sticks or SD cards, FM radio stations, etc. So, they are called differently.
Personal device is usually called плеер for generic devices or if brand is widely known, you can use brand name instead (e.g. айпод or iPod, волкман or Walkman, etc.).
Stationary devices that one usually uses at home and rarely movable - are called музыкальный центр.
Portable devices that can be brought with you can be called магнитола (may be there are more popular names for them however). However for car audio systems it is mostly автомагнитола.
Last kind has sub-classes like CD-магнитола (си-ди магнитола), MP3-магнитола (эм-пэ-три магнитола) based on primary source of audio data.
As for сотовый - it's a cellphone, not an audio playing device (at least it is not its main function). Of course, most cellphones can play audio nowadays.
As a bottom line, 50 years ago there were only 3 sources of music - radio, turn tables and tape recorders and one could emphasize that they listened to recorded music vs. radio. 30 years ago personal players emerged and one could emphasize that they listened the music using personal player vs some big audio device vs radio.
Nowadays there are so many sources of music that the distinction is between "Слушал радио", "Слушал альбом Нирваны", "Слушал сборник хитов", etc. - in other words "I could not influence the song selection" vs. "I selected the music myself".
So, магнитофон is not archaic, it become more specialized.