By “modern”, I mean “Wasn’t simply a video of the band playing their instruments”, but more like had a story on it, had the band members doing something while singing/playing, or was basically a totally unrelated film set to the song?
I’m also not counting excerpts from musicals - while they’re definitely recognisably “music videos”, they’re not quite the sort of thing you’d see on MTV or Rage or VH1.
I want to say The Beatles - the video for Rain: The Beatles Rain - YouTube
Ricky Nelson’s Traveling Man from 1961 is an early example, starting with the second verse, about a minute-and-a-half in.
There were musical shorts going as far back as the invention of sound movies that fit all the definitions of a music video.
That, of course, would be 1920’s-style Music Videos.
Tony Bennett claims his 1954 short film where he sings “Stranger in Paradise” was the first music video. Not surprisingly, other artists also claim to have firsts.
I kind of agree with RealityChuck about Ricky Nelson, inasmuch as I think it was the first music video intended specifically to go on TV.
Oooooh I wanna see AND hear that Beatles viddy. Prairie poop!
From about the same era: The Kinks did a video for “Dead End Street”: https://www.youtube.com/watch?v=zhj0vF1KN1k