I think I tend to agree with you: I like both, but consider the "fast zombie" to be a totally different animal.
I try to use the term "ghoul" rather than "zombie" for both, to distinguish them from the very different and all but completely out-of-fashion "Hollywood Voodoo" style zombie movie (see White Zombie, I Walked with a Zombie, Plan 9 from Outer Space, or King of the Zombies for some examples, or Phantasm for one of the last modern examples I can think of in this genre, with diminutive, hooded hobgoblin zombies shrunken into hideous dwarfs to serve as a slave labor force on an alien planet for a unique twist on the basic concept!) For a "zombie" movie in that sense, I kind of expect a "zombie master" mad scientist working for some sort of sinister organization - typically Nazis, Ruritanians, or aliens - who use some sort of super-science - probably serums, radiation, or hypnotism - to create a race of zombie supermen to take over the world, unless they are stopped by some plucky investigator protagonists. Unless the movie is more specifically about a Hollywood-style Voodoo.... The Hollywood Voodoo zombie movie's hayday seems to have been the '30s and '40s, and a survival through the early Cold War, for obvious metaphorical reasons.
I think the slow "ghouls" actually have a pretty long tradition in sci-fi, actually predating the fad of Hollywood Voodoo "zombies"... one of the more famous early examples would be Night of the Living Dead (1968), but that movie clearly took many of its cues from that great little Vincent Price version of the story "I Am Legend", The Last Man on Earth (1964). But, I think The Day of the Triffids (1962) pretty much codifies this style of movie (it's a modern "zombie movie" in almost every way except that walking plants stand in for ghouls), and Carnival of Souls (1962) pretty much establishes the low-budget look and feel for what ghouls look and act like, and I can't help thinking that there's a healthy dose of this movie in the ancestry of the "Living Dead" movies.
Interestingly to me, both Carnival of Souls and The Last Man on Earth play their ghouls as intelligent creatures, playing psychological mind games with their victims, and even laughing at or talking to their victims! This seems to be a lost element of this "trope", and I assumed for a long time the brain-dead "vegetable" aspect of the modern "zombie movie" ghouls came almost literally from Day of the Triffids: whatever the case, Night of the Living Dead seemed to cement the notion of slow-moving, inevitable ghouls into the imaginations of movie-goers and movie-makers in the years to come.
But, the origins of the post-apocalyptic "zombie movie" go back a little ways before the mid 1960s: there was a fad for post-apocalyptic movies in the '50s to the early '60s as well, with many of the now familiar tropes of desperate survival in the face of a collapsing civilization appearing in post-atomic-war movies like Panic! in the Year Zero (1962) or Five (1950), and small ad-hoc groups of bickering survivors being besieged by slow-moving and awkward atomic mutants in movies like The Day the World Ended (1955) or World Without End (1956)... it seems that the "Living Dead" movie series owes a bit of its ancestry to these movies as well, substituting ghouls for mutants - in the case of Night of the Living Dead, those ghouls seem to have been caused by radiation or some other contamination leaking from a downed military satellite, all but explicitly making them close cousins, if not twins, to the atomic mutants of the previous decade! These post-atomic movies would enjoy occasional revivals into the '70s, '80s, and byond, with The Day After (1983), Damnation Alley (1977), and hosts of others, including the '80s fad of movies ripping off Mad Max, which gave a high-speed action flavor to this particularly cynical branch of the "zombie movie" extended family!)
But, perhaps the grand-daddy of them all was a movie inspired by no less than H.G. Wells, from a surprisingly pre-atomic era of science fiction: Things to Come (1936)! Amazingly, nearly all of the components of the modern "zombie movie" are there, in a not-quite-finished form: a story set in a (then) near-future 1940s, predicting a second world war which bombs the whole world back into a virtual feudal condition, followed by the release of a biological weapon called the "Wandering Sickness" that turns its victims into mute, mindless, slowly shambling, wandering "zombies" that pursue terrified survivors with the threat of contamination through eerily ruined and deserted cities, while petty self-made feudal warlords organize rag-tag armies to raid and conquer weaker communities, and a secretive organization of more technologically advanced survivors are at work on rebuilding a utopian new world, one village at a time.... This story seems suspiciously like the template that The Walking Dead would be built from, and no doubt left its mark on many an earlier post-apocalyptic movie and "zombie movie"! (Some familiar post-apocalytpic horror themes run through H.G. Wells' earlier works, too, such as The War of the Worlds, where a conversation with a military deserter reveals his scheme to commandeer a Martian war machine and turn its conquering might on his fellow human beings to become a master over the human survivors, and even in the plight of the Morlocks from The Time Machine, driven into cannibal madness by centuries of post-apocalyptic exploitation by a decadent and likewise deteriorated Eloi leisure class.....)
So, I think those slow-moving zombies are a very old tradition in sci-fi/horror, one that seems to be pretty tightly connected to apocalyptic settings and a fear of contamination, and where as much of the story's central horrors and conflicts come from the fall of civilization and a descent into animal barbarism as much as anything else: much like the ghouls are theatening to contaminate the protagonists and turn them into literal walking dead, the story's assorted savage warlords, cult leaders, and sociopaths threaten to turn the story's protagonists into a different sort of walking dead: inhuman savages who have lost the last of their humanity to desperate barbarism in the name of survival....
And that brings me to those "fast zombies", which seem to be a comparatively recent phenomenon, and, as your commentary hints, one that seems to derive its horror from slightly different stuff: rather than a slowly encroaching dread and a more metaphorical theme of contamination and loss of humanity on various levels, these "fast zombie" movies seem to (literally) cut to the chase, running on high-speed chases and the threat of being quickly consumed by a menace that seems to have a lot more in common with a primordial fear of being chased and eaten by faster, stronger, hungrier predators, than it does with the H.G. Wells' "wandering sickness" tradition.
The origins here seem murky to me... they seem to borrow at least bit from that cynical, high-speed, car chase action-and-surival heart of those '70s and '80s Mad Max style post-apocallyptic action/adventure movies, and Carnival of Souls does seem to be an early example of surprisingly nimble ghouls playfully chasing their protagonist through a deserted carnival pavillion, and one can't help suspect some of this sort of movie's ancestry to be traced to exploitation fare like Cannibal Holocaust or The Hills Have Eyes or Death Line (a 1972 British movie about a family of mute, cannibal hillbillies dwelling in the London underground, chasing victims through empty train stations to devour them - an obvious 1970s-modern take on the rather ancient legend of Sawney Beane), and they possibly share some conceptual DNA with movies like Aliens and Starship Troopers (subsituting the running dead for fast-moving alien xenomorphs and bugs!), but I don't think I can reall anything much closer than that appearing in living dead movies until the 1990s or 2000s, with e.g. Ghosts of Mars (2001), The Descent (2005), and 28 Days Later (2002). These sorts of movies sometimes seem to draw on a few of the same tropes as their slower-paced cousins, such as ad-hoc groups of survivors who can't seem to get civilization right between themselves, or conflicts with barbaric warlords and the like, but I think these elements tend to be downplayed a bit compared to the slower-moving examples, or perhaps altered in subtle ways (conflicts with those post-apocalyptic warlords seem to take on more of a revenge-horror type quality, than an exploration of some idea of being contaminated by savagery....) I tend to assume slow-moving ghouls for RPGs and other games.
Oh, and then, there's the "Deadite": a distinctly different breed of Living Dead "zombie", the Deadite (named by one of its earliest appearances, in the Evil Dead movies) is animated by demonic forces, and takes on a distinctly demonic or alien appearance: bulging eyes, distorted faces, claws and horns, the power to levitate and cast spells, use tools and weapons, and speak in hellish voices and taunt their victims. I think the Deadite might be rather under-represented in gaming!