Kynsa has a good breakdown of what each version is, I’ll try to focus on things more from a story point of view. In summary though, what I think you want is actually the Retail version of the game and to explore what Time Walking has to offer.
WoW was never really heavy handed with its narrative, instead choosing to deliver it loosely through quest and literature scatterred throughout the world. I wouldn’t say there’s any real story in the game until WotLK, which introduced an overarching plotline with Arthas that cultimates in fighting him at Ice Crown Citadel. From this point on WoW started to lean more and more into an expansion sized plot line that was increasingly present as time goes on.
So if you don’t care at all about gameplay and just want story, that automatically excludes…
- Classic Era - This is the Vanilla version of the game more or less as it was from 2004 to 2006. The story is mostly all just self-discovery and quest text driven.
- Classic HC - The exact same as above except that if you die, you stay dead.
- Classic Season of Discovery - A remix, if you will, of Classic Era but there’s a whole bunch of new gameplay stuff you don’t care about. There’s new raids but they don’t contain any story whatsoever, they’re just content to do.
So in the Classic realm (which is the forum you’re posting in, by the way), there’s one more entry, which is Cataclysm Classic. This comes immediately after WotLK and focuses on the story of Deathwing and events that were created to support that. Like WotLK, there’s an overarching plot line and I’d say it’s a little more story driven than anything previous, but it’s still at a cross over time. If you’re looking for a blend of Classic and Narrative, this is probably your best bet.
Past that, you’ve got Retail. If story and narrative is your focus, I think this is actually your best bet. In retail, you can go back and experience every expansion the game has had and play through it in a semi-self contained mode via Time Walking. For example, if you’re Alliance, you go find Chromie in the Stormwind park and talk to her. She’ll allow you to select a time period and then you just go play that. You can level multiple characters and take them through time periods to keep things more cohesive. This also lets you go through all of the expansions that came after Cata, which start to have more and more story.
In particular, Legion, Battle for Azeroth, and Shadowlands were all very heavy narrative driven experiences. I’m not much for story myself but I thought those expansions did a pretty good job of creating a narrative and presenting it. It’s worth noting here as well that your $15 per month subscription fee grants you access to all of this. You get everything I’ve talked about and that Kynsa lists above as well as everything Retail has to offer up to and excluding the most recent expansion. Currently this is Dragonflight but if you wait a week or two that will be The War Within, so you’ll get all the Dragonflight content too.
The last thing I want to throw out here is that if you’re interested in what happened post Warcraft 3 specifically, you’re actually much better off looking up a lore video. I did this once at the start of Warlords of Draenor because I was thoroughly confused as to what’s going on. It was an hour long video and it did a pretty good job of explaining things to a noob like me. I remember very little of it, but it’s there 
There’s also another extremely good video about the Defias Brotherhood floating around which was awesome to watch. I never actually realized that there was a whole deep story behind me running back and forth between Westfall and the nearby towns.
Anyway, I hope that helps. Good luck!