I think Shinji Mikami had a huge part in his presence, too
You might be more right than even you think. When Kamiya saw the DMC3 Dante incarnation, he commented that it still wasn't wild enough, and kept giving outlandish suggestions for the character. So it's pretty safe to say that if Mikami hadn't been involved, the game would have wound up with a Bayonetta level of batshit nutty Dante, with only Kamiya at the reins.
On topic, Itsuno always made it clear in interviews that with DMC3 he wanted to make a fresh new start for the series, and that included Dante as well. And considering the fact that with that game the character got to be at its most complex, even to this day, I'd say it's one of those rare cases where the original is surpassed.
Not that DMC1 Dante is a bad character or anything, but it's not like he goes through as much of an arc or development. That version is also tainted by bad voice acting, like every character in DMC1 for that matter, which doesn't help.
4 is another story, that Dante was fun to watch but his almost total lack of serious moments didn't sit well with a lot of people, understandably. Those were very much present in 1 and 3 (of course without descending into deadpan lifeless boredom of DMC2 memory), but at the same time the game was about Nero and Dante was relegated to a side character.
Yet 5 features a Dante that's the closest he's ever been to his 1 incarnation, only better acted and featuring the more lighthearted, boss-humiliating, confident taunting that the fans have loved ever since 3.
As for the anime, I honestly don't know about its development and therefore I'm unaware of how much of a say Itsuno had in that one, but everyone knows the general consensus about that version, and the headcanon constructed around it to try and justify it.
