"Carrier" in this context(Net Neutrality) is short for telecommunications carrier, which are regulated by the FCC in the United States[1].
The classification is a legal one. It was refined by the Telecommunications Act of 1996[2]
AWS/Azure/GCE are neither regulated by the FCC nor classified as telecommunications carriers as they do not sell broadband access.
Simply owning you own "backbone" does not make you a carrier either. There are plenty of large corporations that own their own backbones that are not carriers. Being a carrier has nothing to do with hosting content either.
[1] https://en.wikipedia.org/wiki/Common_carrier
[2[ https://en.wikipedia.org/wiki/Telecommunications_Act_of_1996