Funny thing is that the only recommended carry item for him is BT. Some carry.
People really need to get over the idea that carries have to build 'carry' items.
If you think back to DotA, different carries were based on different sorts of item builds because different sorts of carries carry in different ways.
Ezreal's a carry and regularly builds very differently from other AD carries. Karth/Cass/Anivia are all very real and very hard carries that don't build any AD items at all. There are even a handful of tanky or straight tank champions that can flat-out carry through a combination of high base damages and good disruption and initiation, although I still wouldn't refer to them as 'hard' carries.
Aatrox may be entirely capable of carrying with one AD item, or even none. It really depends on how his kit works out and all that jazz.
Ezreal is not a carry as he has no innate scaling. He's a ranged skirmisher/assassin. As far as ap casters- they are not carries. They are generally assassins/supports/initiators (the ones in your example are assassins with the exception of Anivia who is mostly an initiator).
A carry is someone who scales very well with items and gets more useful later on, with a full or almost full item build. Ezreal (and the ap casters you mentioned) are the exact opposite- they are at their strongest in the early- and mid-game and fall-off drastically near the end-game.
An example of a carry would be Kog'maw- terrifying with a full build and very underwhelming with 1 or 2 items.