Spectre and Meltdown are two different exploits. In this case, the paper is talking about Spectre and they did have a POC that worked on all three major process

Spectre and Meltdown are two different exploits. In this case, the paper is talking about Spectre and they did have a POC that worked on all three major processor families (except for deterministic execution engines in microcontrollers and low end CPUs from ARM as well as a few early Atoms from Intel). AMD is saying that Meltdown, which at first glance seems like the most serious one, doesn't effect their processors.