Off-hand, have you watched the Darker than Black series?

Amatsuki mainly the first but kinda elements of the 2nd too.

Anyone have anything else?

Isekai de “Kuro no Iyashi Te” tte Yobarete
Ore to Kawazu-san no Isekai Hourouki
Death March kara Hajimaru Isekai Kyousoukyoku
Only Sense Online

Perhaps more along the Re:Monster type, but for manga with a reincarnated mc:
Hachinan tte sore wa nai Deshou
Mushoku Tensei: Isekai Ittara Honki Dasu
E? Heibon Desu yo?
Konjiki no Moji Tsukai
Spirit Migration

Any others?

Quote from Minase Kurika

Isekai de “Kuro no Iyashi Te†tte Yobarete
Ore to Kawazu-san no Isekai Hourouki
Death March kara Hajimaru Isekai Kyousoukyoku
Only Sense Online

Perhaps more along the Re:Monster type, but for manga with a reincarnated mc:
Hachinan tte sore wa nai Deshou
Mushoku Tensei: Isekai Ittara Honki Dasu ...

Ya, sadly, I've already found all of those. It is strange how hard it is to find many in the Genre. Is it really that new of a concept?

Um, most of my suggestions are short [I'll label them accordingly, as chs scanlated]... Unless you can read Japanese. But I'll give them anyway, because of a seemingly lack of postings:

Category one:
Tensei Shitara Slime Datta Ken A type of reincarnation into a RPG game. But he's a "slime". One of the "lowest mob monster". Except, he's like a sage slime. [9 chs out]
Gaitou Alice Um, there's no leveling or stuff, but guys complete against each other by collecting "dark feelings", for the right to get what their heart desires. [9 chs]
Alice in Borderland Ok, so some friends who feel the world is not offering all it could to them, stay up, see some fireworks, wake up to a world that "stuck in time", and get thrown into "games" which they must win to live. But they could die while participating in the games as well. [44 ch]
Good Night World Same mangaka as Sekai Oni. This one is his newer work, so there's not much out. [16 chs]
Kumo desu ga, nani ka? Some girl this time, gets reincarnated into a spider. Apparently, she levels up into a very hot humaniod spider who gets drunk easily. [3chs out]
Wizard's Soul Holy War of Love It's a fantasy, otome type game, where the girl levels up as she gets closer to her captor target... I'm kidding (I'm sorry.. I just wanted to have a laugh). It's actually a card game type manga. [7 ch] No fantasy, but the card game rules are from the mangaka's imagination (I think it said somewhere that the mangaka based in off of Magic?) Might not be the type of game you're looking for though.

(No, it wasn't on purpose that two of those mangas have Alice in them being based off Alice in wonderland...)

Category two:
Sekai Oni The art for this one is kinda... pencil like? But, the plots pretty good. Give it like 10 chs though. It kinda just drops you in a mess of things at first. [Completed 101 ch]
Ptolemy's Singularity If I remember correctly, which probably means I don't know what I'm talking about... [41 chs]
Ore to Kawazu-san no Isekai Hourouki Short, only 16 chs out. So some guy gets transferred to another world because an old sage magician is dying and the sage wanted to past down his wisdom to someone with high mana (apparently modern day Japanese people have really good stats). I'm putting this one down here because he introduces the internet to this "new world". Although it's not exactly what you wanted as "cross dimensional cultural exchange".