"No Intro" does not refer to the intro cinematics of the games themselves. Back in the day, groups that used to release cracks and roms would often add little logos to the beginning of the rom or cracked game. So like, you might download game and when you fired it up, it would say, "Brought to you by Ice Devil's Hacking Collective. Find More Great Cracks on our BBS at (555) 555-5555." The no intro sets were just put together as collections of games and roms that were unchanged from the originals and didn't have those intro tags. Putting those little tags on PC games and roms is not common these days, but the "No Intro" collections are still considered the gold standard. All the roms on Vimm were pulled from No Intro sets, for example. The name is a bit of a holdover.
Here's a couple examples I found Googling around real quick:
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Back in the 90s, you would sometimes download a NES rom from a random geocities site, and when you loaded it into nesticle, something like the above would pop up before the title screen or sometimes it would even be integrated into the title screen, or written into something like the Mario 2 or Zelda story screens.