No worries!
That does clear things up, thank you for that. I will admit that a lot of the sorting out of modern(-ity/-ism)/postmodern(-ity/ism) can get a bit confusing for me when the various ideas that get produced from these tensions flow from the academy to popular culture and discourse (both of which often misread or selectively Not Read the other to much annoyance and then keep on trucking with that mis/not-reading). In my own case just now, I will admit to carrying over Mohaghegh's annoyance at (what I understand to be) a Western Academic postmodern reading/philosophical practice that either gets completely detached from the object it reads out of a logic loop of suspicion and paranoia, or it clothes itself in a pseudoscientific guise and asserts a theoretical framework as The One True Theory (in his own words, "masochistic skepticism or sadistic truth"), as well as one of Mark Fisher's frustrations of how -pop culturally- there appears to no longer be a shared space or overlap between the avant-garde and the mainstream that can open up and build new worlds of consciousness (for him, the understanding of modernity as pointing to an inherent, shared reality would be more crucial to the project of building (class) consciousness which he saw as vital both politically and philosophically). Ultimately though, and this is partially thanks to the permissions postmodernity can give, I don't so much view these and many other philosophical, hermeneutic, and theoretical concepts and devices as totalizing frameworks and more as tools in a toolbox. We're working with, to borrow a lovely phrase from twitter, Lego Philosophy, not IKEA Philosophy; if the thing you're building with legos doesn't do or be what you intend it to do or be, then you either reassemble the legos or you bring in new legos until you can build that thing in question, and often in the process the nature of the bricks you're building with can suggest and produce something unexpected as you're going along.
To speak more to hauntology...hauntology is a bit of a weird term, partially because I understand it tends to be memified to mean just about anything with a sort of wistful, melancholic-nostalgic quality (less hauntology like biology and more hauntography like biography; the former being the study of the process of life, the latter being a document of a life as it was lived). My own understanding of hauntology comes from Mark Fisher, which I understand he copped from Derrida (which Freeman's use of it seems to stem more directly from, from what I can pick up on a quick search). If Derrida's concept of hauntology is more a space or phenomenon where history interacts with the present, where the dead "call for a different future" than the one we are able to produce, Fisher's application is more of a study/phenomenon of moments/media where we not only see the ghostly repetition of these lost future possibilities (futures that never got to materialize), but also a critique/commentary on that ghostly repitition of the past's affect upon the present to either open up new (sonic) worlds or make eerie the present worlds we occupy. The two major musical acts that Fisher tends to cite in this are Burial and Leyland Kirby/The Caretaker. With Burial, the hauntology comes from his music which acts as rave music for the memories of a rave long since gone, a sort of "downcast euphoria" as Kode9 put it, a 3 AM wandering through faded graffiti streets and angelic visitations in a McDonalds. With The Caretaker, there is the decaying repetition of memories of home, a crackle of 50's music that repeats on seemingly inconsequential phrase loops, the playing out of homelikeness that skips into sudden absences, the eerieness of suddenly finding yourself in a house alone, not so much wistful nostalgia but the shock of a sudden melancholic tenderness and vulnerability.
(A good book to dig into Fisher's use of hauntology is his own "Ghosts Of My Life: Writings on Depression, Hauntology, and Lost Futures" if you want to dig into that as a starter. Online, I'm pretty sure you can also find the full transcript of Fisher's interview with Burial, which is a fascinating read in its own right as well as a discussion of this concept both from the theorizing and the doing sides of it.)
At least in my essay, I was putting the articulation of hauntology through music in a group of cases of a weirding of culture through specific readings of media that I've noticed, alongside the mathematical crystallizations of poetry as it relates to Stephen Jonas and the suggestion by Jean Epstein that film/cinema has an intelligence, is "an experimental apparatus that constructs, that is to say, thinks an image of the universe whose reality is predetermined by the structure of its plasmatic mechanism" (from "The Intelligence of a Machine"). When I referenced hauntology, to be honest I had a hard time imagining what or how video games COULD be hauntological...for me, hauntology is deeply tied to space (my listening to Burial's "NYC" on a plane ride at night and feeling a deep, oceanic bliss within that song, or stepping out of my room when no one else was home while The Caretaker's "An Empty Bliss Beyond This World" plays from my laptop, turning isolation into a bridge of ghostly contemplation).
As unfortunate as it is to need to reference it, right now we have the best example of how hauntology fails to manifest in the positive/new-making aspects as it did in Burial and projects like him: Cyberpunk 2077. Not ONLY is it a sort of remastering/remake of an old cyberpunk TTRPG property that does little to update/reinvograte/reweird it (and in fact arguably just makes a duller echo of its source material) and even less to project ANY new horizon of science-fictional futuremaking (cyberpunk has always had multiple strands of weirdness, from the nihilating punk "no-future" Mad Max strain to cyborg transhumanism to Gnostic Cyberspace Transcendentalism), but EVEN AS A GAME it does nothing to make anything new of the same formula that Every AAA Prestige Title Has Ever Done, INCLUDING The Witcher 3 -here's a vast, open world, filled with numerous glitches and lists of tasks and items to fill out across the entirety of this map, a series of disconnected minutiae dumped into a desert and scattered so far that hopefully no one notices how hollow it is- except this time one of the glitches is your avatar's bepis clips through their pants. The Witcher 3 itself feels less to me like a using the Fantasy Genre to interact with cultural forms and project an imagined past that can represent future longings (see Moorcock's Elric saga and how its edgier, 60's psychedelia infused countercultural current is projecting itself into an imagined Past Beyond Our Past, a History Before Written History that we are implied to be the inheritors/descendants of), and more like a "Historically Realistic Fantasy" that has less to do with either "History" or "Realism" and more to do with Prestige TV. In Fisher's sense of the word, I wouldn't say The Witcher 3 is hauntological at all, in fact it and Cyberpunk 2077 are the very thing hauntology is meant to critique, the repetition of an aesthetic signifier of the past evacuated of any of its radical potential and played out on loop with no escape or difference or detournment.
So then, that leaves a really pressing question; what the hell video game IS hauntological? Maybe Disco Elysium? Could something like Shadow Hearts be hauntological, with its urban fantasy epic taking place in an occulted backdrop of the eve of World War I as the immortal philosopher Roger Bacon attempts to summon an Eldritch God to Earth? Is Tabletop Simulator hauntological (or potentially hauntological) as a virtual simulation of board games (or is it too Baudrillardian)? Is hauntology possible in the various god games and factory simulators (SimCity in all its iterations, Factorio, Dwarf Fortress, Creatures)? Are SHMUPS possibly hauntological, and if so is the kind of fantastic mythographical slice-of-life of Touhou more apt of a subject of hauntological study, or the more existentialist and melancholic examples of Radiant Silvergun and R-Type Final as they flirt with the end of all of knowable existence? Is P.T. an example of hauntology in video games, the use of horror game's conventions into the construction of a discomfiting puzzle box, the leaking of Silent Hill into the no-place of the suburban home and the game console not through Pyramid Head knockoffs and rust but through the transformation of the console's various microphones and sensors into an extended network of synapses to put into fight-or-flight duress and paranoiac investigation (sifting through the code you find that the ghost is always behind you from the second you enter the house)? Is the defeat and yearning of the Zenkyoto student riot movements in Japan so thoroughly buried and abstracted out that hauntology becomes either a frantic archaeological excercise or flat out impossible (at least as video games are concerned)? Or are we looking at the wrong place as far as "official" video games are concerned; is hauntology instead in the mass of doujin games and bootlegs and mods (Doom .wads, Tamriel Reborn, the entire apparatus of RPG Maker and its many iterations)?
For now, it's hard for me to say, but these are the thought sketches I would have to offer to answer some of your questions.