Goblin: grim reaper and his morning drama
Business proposal: fake drama that mirrors the real drama
Reply 1997: mom calls drama writer to change the story

So Not Worth It
I hesitate to recommend this because it’s not everyone’s cup of tea, it’s mostly slapstick humour and has some cringe acting. But it’s earnest and I’m a sucker for dorm life/ shared living stories

One of the international students Minnie is obsessed with Kdramas, Crash Landing On You is among her favourites

Grandpa in Business Proposal and Scarlett in Search: WWW were drama lovers for the dramas within a drama Be Strong, Geum-hee and What’s Wrong With My Mother-In-Law?

Oooooh I was racking my brain for a minute.

I would go with Search: WWW! The drama scenes were funny and added a fun layer to the secondary couple's story.

Another vote for Grandpa in Business Proposal - especially when he is commenting on his drama, "like any rich family would meet their inlaws in a pork cutlet franchise." Okay, Mr. Pineapple Socks... cause we all expect unrelenting realism from a drama.
I just love his character anyway, I mean, you know the whole time that he'll end up accepting Ha-Ri, but it's still hilarious to watch him chasing Tae-Moo around trying to beat him with a cane.

There was a scene in Goblin with the Grim Reaper watching a drama in a restaurant. It was pretty funny.

Reply 1994 had a long scene about Sandglass with what was probably a very accurate depiction of its impact right down to the Choi Min-soo vs Lee Jung-jae arguments.

Shin Hari's parents and Go Foods Haraboji in Business Proposal. Shin Guem-hui worn the hearts of all the social classes

And Kiss Sixth Sense with the Haroo story. What better way to make a tetchy sad conversation you wouldn't necessarily have open to discussing? Haroo provided them a means to closure.

Waikiki family.
Scarlet (Search WWW).

The North Korean soldier in Crash Landing. Fanboy’s dream come true when he actually meets the FL of his kdrama!

was there not a scene where "Yoo Na" checks out Park Seo Joon in Kill Me Heal Me and asks whether he is a drama actor.

Ji Sung's father in Protect the Boss

Is it too obvious to go for the confession scene in Be melodramatic when they are walking and act out two or three versions of it?

There’s Dramaworld, a Viki production that was super-meta as the kdrama fan heroine gets transported to being IN a drama:
https://en.wikipedia.org/wiki/Dramaworld

Where's Muggy? Muggy should be here
@ndlessjoie

In Doom at your service, she watch dramas to find out how to fall in love with a supernatural beeing.

I know this is my own request, but I thought of another one!

In Vincenzo, doesn't Joon-woo watch a fake drama when he's hiding out in the countryside mansion? Before he gets poisoned by the bottled water? I don't remember what happens in the drama, but I remember it being VERY dramatic and I think it was referencing one real-life drama in particular. Does anyone know more about this one?

Also Because This is Our First Life... Ji-Ho is a drama writer, and the whole first scene is watching her cram a list of crazy PPL into the script, rolling her eyes the whole time. Then later on there's the scene where the producers take her drama script and hand it to the writer, who proposes all kinds of nutty tropes to add into it to "improve" the story...
This whole series focuses on her being able to tell a "real" story, and ends with you realizing that the series itself was her "real" story (about herself). Essentially telling you that you have already been watching her real-life drama... although nobody in the series watches dramas, they watch soccer games.

We saw a made-up movie in Mystic Pop-up Bar - Gout Lovers .

In Vincenzo, one of the characters made fun of another for looking like Park Saeroyi because of hair. I didn't watch Itaewon Class, but I knew exactly what it referenced!

In My Girlfriend is a Gumiho, there was an imagination sequence from a side character that referenced the hit drama at the time, Giant.

Scarlet in WWW Search!