Superhero RPF is a fanfic about fanficers so it has a lot of fanfics in it. Most just referenced by their title and summaries, except For Asgardia!, by a guy going by the username Sassgardian, from which there are excerpts all around the fic. It gets him accused of Draco In Leather Pantsing for his heroic interpretation of Loki, and when his identity comes out, he is in fact Loki writing Self-Insert Fic.
The Skyhold Academy Yearbook series has several examples in the form of various characters writing Real-Person Fic about themselves and each other. If Varric Tethras isn't the one writing it, he's usually the one reading it out loud to other characters (and thus to the reader).
"You wanna plop down and watch another episode with me, Misato? Shinji and I have been watching it lately. It's like someone heard about NERV, took a lot of drugs, and then tried to make it a comedy-drama-angst-romance with an ensemble cast of traumatized lunatics. It's a cheesy giant robot show, but a lot of fun. Everyone in it is pretty hot, too. Especially that Akram guy. Yow."
"They've got Gekiganger X down at the movie theatre today."
The Ikaris: The "Super Gendo Super Show", created by the PR department to make people more aware about Commander Ikari. It is like Evangelion on crack with a healthy helping of Super Robot Genre and tokusatsu tropes. They are making lunchboxes, too.
Futari wa Hearty Pure. The Cures (and also most people at their school) are big fans of the show, and Lapinyuu becomes part of the cast, playing a Shiny Luminous parody.
The two shows that helped Chloe and Chiara's father Theodore become a trampoline fetishist, Bounce and Thud, also count.
Rosalina loves the show "Comet Academy", which comes on once every 86 years. Peach and Daisy find it utterly boring.
Rosalina: 12 Lumas are sent on the Academy Comet, which is the host of the Comet Academy! During the 86 years it takes for the comet to come back, the Lumas are training by eating the most Star Bits they can, so once in competition, they can transform into the most formidable things!
In My Huntsman Academia, Izuku has an entire collection of comics titled, "The Adventures of the All Mighty Huntsman", which are a loose adaptation of Toshinori's various adventures. There's also a TV show based on the comics, in which Toshinori's voice is impersonated by a guy named Chris Sabat.
Examples of type 3 (the show is a plot point)
In Equestria: A History Revealed, the Lemony Narrator writing the fic-essay decides to go out drinking and just copies a large block of test from a book on a soldier's recounting of a significant battle in the history of the Equestrian Civil War into the fic. Notable in that the copied text proves to be vastly different in tone than the rest of the tone of the fic, probably due to its In-Universe copied nature, and is even written in a more high quality fashion than the rest of the essay-fic.
In Let Us Be Your Poison, Violet and Daisy is a propaganda-based kid's Edutainment Show featuring puppets teaching kid's how to be virtuous. It runs on Atlas' state television. Each episode ends in the duo singing about respecting family. Weiss and her brother Whitley used to watch it as kids.
Examples of type 4 (the show is eerily similar to the main show)