Diamond was the worst entry in series history. I'd suggest trying Black or White if you visit the series again. If not, I wouldn't blame you though. Diamond had a lot of things against it.

FF13's story was at least interesting enough to get me through the game. 12's on the other was so bland, cliche, and entirely predictable (and certainly NOT subtle and nuanced as many suggest) that I couldn't even finish the game despite a decent battle system.

I don't really think of Pokemon as a typical JRPG though, it's nothing like anything else out there. (other than Pokemon clones) It is a category of its own.

So, yes. A JRPG definitely needs a good story. It is the driving force behind the game.

A great story compensate for lacking gameplay. You can often do away with story with great gameplay.

Looks at top selling JRPG's... Pokemon, Monster Hunter, Dragon Quest, and Final Fantasy,
3/4 of those are generally heavily gameplay based... :D. Obviously this thread isn't representative of the general population.

Great story can cover for lack of substantial gameplay (FF6). (to some extent)
Having a good gameplay, can cover for a bad, and generally this has been the case for years and years.
it honestly depends on the person. Some people will find certain things more bothersome than others. Some plot inconsistincies will bother certain people more. 

XIII-2's plot definetly has more issues. Not the same issues as XIII(which I'll defend). And I still enjoyed it, as it's light-hearted moments made me laugh a lot.

WHOA WHOA WHOA, did anyone just say TWEWY's battle system was mediocre?

Yo dude, i'm really happy for you and imma let you finish but twewy had one of the best battle systems of all time, OF ALL TIME.

Oh and the persona series by far have the best stories in jrpgs.

Press start, then select = cutscene skipped. Durrrr!

The cutscenes are worth it, though, just because Noel is voiced by Jason Marsden from Boy Meets World. Mind blown!

Usually yes, unless you can pull off mechanics so great that nobody gives two shits that the story isn't worth a damn. (I'm talking about you Pokemon)

Pokemon and Games like Demon's Souls say no. Gameplay can always drive the experience and keep you engaged.

I spent 40 hours playing FFX. I spent 200 hours playing Pokemon HeartGold. Guess which one had the better story?

Hint: It wasn't the one with a mute protagonist.

I've always played JRPGs for the varying and engrossing combat systems, the combat in an RPG is something WRPGs have only started to greatly experiment with now as can be seen with Mass Effect, Witcher 2 and Amalur. Funnily enough they're some of the few WRPGs I greatly enjoy. Fable is another but that's more of a guilty pleasure.

FF12 is the only JRPG which comes to mind with good dialogue and adequate storytelling. That and the FF Tactics PSP got re written in a similar style to FF12s dialogue I believe. Lost Odyssey also has well written segments but these are ancillary to the main quest from what I remember.

I can't pick up Chrono Trigger today without cringing at the story yet it's a classic, that's the beauty of it though. The combat and game design in general is good enough to keep me immersed and captivated for dozens of hours. Personally I believe that's a greater achievement than a well written story in a videogame, after all videogames are for playing but that doesn't mean we should not have both good story telling and gameplay. But just as an example, Super Robot Wars is one of my favorite SRPG series ever and yet I don't understand a lick of Japanese, so clearly story is a non issue in regards to my enjoyment of the games.

As for FF13-2 ,Lightning and Caius had rather well written dialogue in FF13-2. Downside was Serah and Noel are in there for the kids to stay interested and sort of spoon feed you everything. Everything else about the game is great. I literally need a guide to 100% this game, struggling collecting all 160 fragments, FF13 however I came close 100%ing without even consulting any guide what so ever. Poor story? When you have 99% of reviewers whining they don't understand the lore in FF13 it's clearly a case of poor story telling and not so much a poor story. The worlds of Cocoon and and Gran Pulse are enriched in captivating lore, problem is the dialogue is what fails to immerse and leads us to refuse to allow ourselves to care. I find most JRPGs in general suffer this same problem. When I said Chrono Trigger makes me cringe, it's the dialogue and not so much the world which has immersed me through other means.

tl;dr no