For my aged opinion;
Assassins are there to take out an enemy carry, either mage or marksmen depending who's more valuable to have removed (such as before a fight or a siege), ideally it doesn't even require a kill, enough damage to force the carry to retreat or die should be plenty of purpose for the assassin. Problem is they're often feast or famine in pvp environments, being frustrating to fight or play as because they can snowball either way. High single target damage and extreme mobility are what define them, I'd include divers as pseudo-assassins since they tend to share both playstyle and builds, summed up as hit & run.
Tanks are the brawlers that disrupt the enemy team by force, being durable with plenty of crowd control are what they're known by; they can be supportive to setup allies combos, or steamrollers to kill enemies who ignore them too long. Like divers, juggernauts I'd put in here for the similar immobile playstyle and build, however their difference is they sacrifice crowd control for damage, making them a more immediate threat but less valuable as a front-liner for squishier allies.
Skirmisher is basically the new bruiser, melees who have a good mix of damage and durability, with either some mobility ability or a crowd control; however they're more like tanks than assassins in terms of playstyle, wanting to stay in longer fights to grind down their opponent as opposed to burn everything to nuke them. A few divers I suppose could fit here, and they're the unviable ones because others can do their jobs better; this and fighters are what most melee's fall into because of the massive ranged advantage in the beginning.
Fighters . . . the outdated terminology to blanket label melee champions in the grey area; having a mix of mobility, damage, durability, and crowd control. They're the flexible picks who can fit into a similar role as someone else without entirely committing to it. Way back in the first couple seasons of league, melee's struggled against mages and marksmen alike (mobility wasn't as common and mages were alternate team carries), so riot started making melees with a mix of tools to fight back; most noticeably irelia and xinzhao, and since then it's been the general design for non-ranged champions that aren't designed as another class.
Mages were the magical powerhouses, limited by mana and cooldowns (until the last couple years) but having more variety than the melee champions for being more focused for something; there's artillery , closecombat (aka battlemages), combo-casters, and utility. Mages in general are immobile but ranged, AOE but delayed from cast time, primarily damage with some utility effect for teamfights. The artillery hardly has any ability more effective at close range (or good selfpeel), and as their name sake, has significantly farther reach for damage. Close combat mages have some durability or sustain to compensate for being close to enemies, usually on lower cooldowns or lower mana costs since they rely on spamming; these are the overlap of mages and skirmishers. Combo-casters are often medium ranged and are the kill-pick mages, generally sacrificing aoe for single target damage, throwing several abilities to quickly burst down their target, but too squishy to stay on the frontline and lack the utility to do much else. Utility mages on the other hand have multiple crowd controls or include some buffs/debuffs in their kits besides some damage, these are the ones picked for support but rarely seen on their own because they're more effective with friends. Mages who are mixed don't have a label like fighters, they're just referred to as mages, lol.
Marksmen have always been the objective (towers and epic monsters) taking champions, together with mages were known for being incidental siege champions, having unavoidable ranged damage that didn't have mana costs yet could attack faster, think skirmishers who sacrificed their durability for range. Like mages they have some variety but not nearly as much, and there's plenty of debate how to slice them for subcategories, but for example here's a discussion I've seen on it ( https://boards.na.leagueoflegends.com/en/c/gameplay-balance/lT3B52ki-in-my-opinion-for-marksmen-to-be-a-healthier-class-riot-should-flesh-out-their-gameplay-niche?comment=0043 ). Anyway back then there was also the dodge mechanic, like critical hits was RNG to occasionally negate an auto-attack, but to get 100% critical hit chance it required building five items; in the class reworks of season six this got changed to only be four items with some being interchangeable, and allowing the extra item slot to be whatever they needed. Critical hits have always been a hot topic because they're self-synergizing damage amplifiers, which at full build have drastically out-scaled mages; and thanks to the extra item they can fit in, can either nuke tanks with last whisper, or sustain through any poke with a bloodthirster. Riot has been trying to subtly (less so in recent years) make marksmen self-sufficient, which is a balancing mistake as long as they're reliant on unavoidable and resourceless damage, that scales higher than anything else without effective countermeasures.
Supports are generally the most diverse, not being a sole class but often the team's punching bag, as they're almost always behind in both gold and levels; however they supply much needed utility about anyone on the team can benefit from. Problem is riot has changed most of their items (over the years) to assist marksmen more than any other class, further exaggerating their problem. Supports are where you also see the most variety of other classes because of their kits and builds; tanks can engage or peel, mages can provide peel or coverfire, fighters can be a distraction and crowd controller, ext. Theoretically almost any champion can be played as such, if you've got great teamwork with your lane partner, non-meta picks blindsiding enemies from the unexpected usage. Supports used to be ward-bots way back when as well; gold income was harder to come by, wards cost gold, there was no such thing as trinkets, and there wasn't a ward cap. I'm glad they gave some useful power to supports, but I'll criticize their lack of preemptive nerfs on both kinds of carries to compensate for supports providing even more team effectiveness.
Now for your ending question: I'd say marksmen should be like ranged skirmishers, they'll kill other champions if left alone long enough, but should be more effective at taking objectives instead of killing enemy champions, however towers/turrets should be buffed accordingly (or several nerfs undone) so it's more obvious why the marksman is on the team again. After all the rift isn't a deathmatch, and used to have more strategy.
that non IE builds will actually be viable on some ADCS more than others.
