A lot of older games liked to confuse difficulty with muscle memory, and skill was rarely a factor. The games where skill is a factor are primarily competitive games, against another opponent, and that skill is the skill that comes with reading your opponent. Games that depend on having high reflexes and co-ordination result into forcing yourself to build muscle memory, which isn't fun or challenging to a player's mental state. Videogames are not an athletic event, they are as far removed from the physical world as possible, with the notable exception of rhythm and sports games utilizing motion control or a special controller altogether. The point of a videogame is to provide mental enjoyment. A good game stimulates your cognitive ability. This comes through things like story and gameplay, however the very notion that story and gameplay have to be entirely separate is silly (although that often tends to be the case with JRPG's, which isn't necessarily a bad thing.)
It can be okay for the gameplay to be crap if the story is good (almost every JRPG ever) for the same reason it can be okay for the story to be crap if the gameplay is good (almost every shooter or WRPG ever). Yes, a video game can have crappy gameplay and still be a good video game. IT HAPPENS. A LOT.
All videogames have some level of skill which is the basic ability to press buttons. For the rest of this post, skill does not refer to button pushing ability unless otherwise stated because I don't consider it a skill in the same way I don't consider your ability to dress yourself a skill. That is called figuring out how to work your game/clothes.
Games can be really skill intensive, but this does not come in the form of rapid button pushing. Because that is retarded. A game like SSX 3 however, involves timing your tricks, setting up massive combos and collecting as many multipliers as you can, and navigating an often treacherous map for a ridiculously high point total. Any one of those things is easy on it's own. You do not need superior reflexes or co-ordination to pull any of this off. What you do need is an understanding of the game's basic mechanics and an ability to link things together, while dealing with the stress and tension that builds up as your stakes raise (messing up the your 3 million point combo nets you nothing) and as subtle variations in how hard you press the analog stick send you on different paths with varying and often difficult terrain. THIS IS NOT A GAME WHERE YOU MEMORIZE A SERIES OF BUTTON PRESSES TO DO WELL IN. It is entirely optional to get this good though, you don't need a platinum medal, and you can place in the top three without getting a million points a run. But if you want a challenge, you can test your raw ability- your grasp on the gameplay and mechanics- by trying to do one massive trick down the entire slope. And it's hard. It isn't unfair. It doesn't require any sort of herculean physical ability. It is your pure mental ability to link together a bunch of different singular skills quickly and under rapidly changing conditions. IT IS A GAME OF MENTAL ABILITY.
Super Smash Bros Melee and Street Fighter demonstrate a different type of skill. You can select one moveset, and using this moveset with it's varying properties you need to challenge and beat a different human being with either the same or a different moveset. It doesn't matter that you can press the A button 500 times per second, because you have to wait for your lag to end after hitting it the first time. Rapid button pressing loses it's value. The value comes in predicting what your opponent is going to do so you can select the move out of your set to punish them, and knowing enough counters and set ups to throw your opponent off balance. It is a game of mental skill in the purest form- one person versus another. Since both players could be the same character if they wanted, the game is fair. THEY ARE GAMES OF MENTAL ABILITY.
Halo, Call of Duty, and most of the online shooter genre also are games of skills. You have terrain, with certain weapons and abilities being dispersed in certain locations. Control of certain vantage points, use of cover and cooperation between teams, and your ability to accurately predict a moving target's position and fire at it are all things the mind, and not the body, has to learn. THEY ARE GAMES OF MENTAL ABILITY.
The only kind of RPG to even simulate this is a strategy RPG. Other RPG's usually can't do this, and the methods of doing so are often restrictive. Is hitting the attack or a magic button repeatedly a skill? No. Is finding an enemy's weakness or devising a strategy to counter an enemy a skill? Yes, but only the first time. Your second playthrough of an RPG can be beaten identically to the first one by just using the same strategies. Granted, in SSX I could probably blaze through it on my second playthrough, but that's because I've developed the proper skills to do so. In an RPG, I'm just copying down the options that work.
I'm talking about JRPG's here for the most part, although WRPG's and JRPG's with an ABS are not far removed from this at all. The same strategies on the same enemies will work the second time, the cleverness needed to discover it the first time is not needed to discover it a second. An SRPG can get away differently. Take for example, Fire Emblem or Final Fantasy Tactics (the latter one a little more than the first, simply because there's more randomization involved in its processes like character retrieval, brave, faith, and such, although it suffers more in terms of balancing.) You have to adapt to changing position and deal with enemies that scale as you do. Furthermore, since a lot of features of your party start randomized, your second playthrough will not be like the first. THESE are RPG's involving mental ability.
So while JRPG's and WRPG's offer levels of mental stimulus in terms of their world, sidequests, weaponry, and story, they simply can't really be called skilled games because their foundation isn't about skill. It's about learning and understanding a mathematical system and using it to get through certain tasks. MMO's have a bit of advantage in that they can release enough new content to provide a constant mental stimulus to a player and so the lack of skill doesn't hit them as hard, but I'll rant about MMO's some other time.
This is where storytelling has become key. This is why a game like Final Fantasy 7, where I can defeat every enemy by doing nothing but pressing attack repeatedly (my attack does 9999 damage, beat only by Knights of the Round and some Limit Breaks because it can fuck the cap by using multiple hits. Magic of any sort has no use besides healing spells because my attack will kill them quicker than five Bahamuts using Megaflare constantly throughout the entire battle. That is not an exaggeration. I got to this point at ten years old.), can still be considered one of the best RPG's of all time. (Part of this was because it was groundbreaking, it was the first 3D RPG to really show off what a 3D RPG was capable of, and I could rant forever about how all your favorite childhood games aren't actually that good but we'll pretend they are for now.) You remember books? I love books. Books are great. They tell a story. They suck you in. Do you have a favorite book? Do you like even one book? They provide absolutely no room for interaction on your part. They stimulate absolutely none of your senses (they strain your eyes, but that doesn't count, movies would be stimulation for your eyes. Reading is just work.) Yet, somehow, a story can suck you in and make absolutely none of that necessary.
Stories supply mental stimulation.
Because you lose a lot of the suspense and surprise after reading a story, it's not as renewable as gameplay. Doesn't matter. It provides a mental stimulus, and that's what you're going for. You are attempting to entertain the brain. If you want a story heavy game, you may not want to use a video game to get it across simply because the interactivity is a hindrance, but it's in no way a problem.
I want this post in no way to get across the idea I don't like RPG's. We make RPG's for fun, it should be a given that this isn't the case. However, it my point that RPG's are not built on a skill-based system and that isn't a bad thing, nor is a story-dominated game a bad thing as it stimulates the brain in the same way as gameplay. Our preferences differ based on what kind of stimuli we take a liking to better, which is why people can take such extreme stances on the subject.
RPG's are built off of tabletop RPG's like Dungeons and Dragons, which were not meant to be games of skill but games of math and imagination that allowed you to do anything. WRPG's keep this feel with their open environments and worlds, and being a little choice heavy while JRPG's typically stray and replace some of the versatility and customization of the tabletop RPG with story. This is probably a result of console limitations (JRPG's are older, they have a foundation in text-based games) but the branch between the two primary RPG types is not a bad thing, it's simply different. However, this replacement does not alienate it enough from the tabletop system to give it a sense of "skill." You are not "skilled" at DnD if you can kill a Balor using a set of spells you know are broken. The point of DnD is not to go monster killilng. The point of a tabletop RPG like DnD is to gain access to a fantastic and imaginative world that's constantly changing based on your actions. This is the foundation of all RPG's.
If you're after different kinds of skilled games look into competitive, strategy, rhythm (to an extent, the "skill" at matching beats tends to fade at higher difficulties and become a game of muscle memory again, the "skill gain" is primarily rhythm training), and (some, they're too similar to want to play too many) sports games.