Nothing indicates that the One Power was used to kill Asmo. It was also reported from a signing that "Brandon refuses to confirm or deny that the One Power was used to kill Asmodean."

Besides that, Asmo might have met Slayer off-screen before the murder. In any case, as your last quote shows, RJ confirmed that all Forsaken know Slayer, and he knows every Forsaken.

That Slayer will appear in ToM and likely in AMoL makes it possible that he will confirm in his PoV that he killed Asmo. RJ wanted to reveal it that way: "He said that if occasion permits in the last book and it seems appropriate, the character doing the deed will probably think to themselves about killing Asmodean."