I suppose I'm similar except my label of choice is simply 'piece (of music)'. In fairness, what I enjoy and listen to runs quite a gamut (Really Slow Motions/Two Steps From Hell to previously mentioned Bach (unashamedly enjoy his stuff over most of his near and not so near contemporaries) to JPop stuff (got a bunch of Maaya Sakamoto anime OP's in mind) to Caravan Palace (however you categorise them)).The second video you posted either makes me feel like in church or Castlevania. I consider it a song. For me I am not picky when it comes to most songs. A wonderful piece like this may have a fancy name or style labeled, but is still a song like this. Concert and symphonic band in school gave me a higher appreciation for classical or symphonic music, though it's not something I listen to the most. More of a jazz and all variety person when it comes to most music.
That said, been a while since I listened to jazz in a bit way. I used to think that I could only really stand big band stuff, but your linked track and that mood of sax music in general is nice to chill to, NGL.