You guys just dont have the same definition of a carry . The one definition deviates from the term "carrying a game", while the other one describes an auto-attacking champion(The last one seems to be Riots definition for the champion notes). Ezreal for example can be a carry, but at the moment he is more often played as a caster(Blue Build).
A 'carry' is a definition from DotA that refers to a champion that is generally weak early game and gets out of control if you let them farm for long enough.
League has no carries that are as hard as the carries in DotA, for which we can all be thankful.
The closest we do have are our 'hard' carries. Champions like Kog'maw and Vayne are good ADC examples of this. Tryndamere, Kayle, Jax, and a handful of other melee champions are also quite good at it. Anivia and Karthus are great examples of champions that carry whole games because of how their abilities work. Singed and Nasus manage it because of some unique mechanics that allow them to be incredibly tanky while dishing out enough damage that they simply can't be ignored - Nasus via arbitrarily high base damage on Siphoning Strike and Singed because of the huge cost efficiency of items like RoA.
Fact is, nearly every archetype outside of support has a champion that can - and often does - function as a hard carry.
(To be fair, there have been points where support champions like Soraka have had enough scaling on something like Starcall to be considered hard carries, and Soraka with Rylai's/Liandry's/Abyssal Scepter is still going to shred your whole team. Haven't tried in a serious game so I can't actually speak to how hard it scales, but it seems like it could qualify.)
'Carrying' a game is entirely different and can be done by anyone through proper play. The purpose of a carry isn't to carry a game, it's to add inevitability to being ahead or the possibility of catching up if you fall behind. If you're ahead and have a hard carry who is also ahead, it takes much more to lose then if you are ahead and do not have a hard carry. If you're behind but can manage to get your hard carry enough farm to be dangerous and protect them well, you can make a comeback even while drastically behind in team gold.