There was a Listener and Speakers. The Listener would give orders to the Speakers, who would make arrangements to carry out the contracts.

Although she doesn't describe herself as such, it seems like Astrid has put herself in a role similar to that of a Speaker. However, since there is no Listener, she gets her information elsewhere. Likely it is from direct contact, as you described, and rumors, like when you hear about Aventus Aretino.

According to the old ways, the sanctuary in Skyrim should not have been doing any contracts. Cicero is confused by this because he is waiting for another Listener, and thinks the rest of the Dark Brotherhood should be doing the same. But in the process of waiting, he's seen everyone around him either die or go missing.

I think he's having trouble reconciling what he believes and what he experiences. He still has hope, but he's seen a nearly overwhelming number of good reasons to give up. Combine that with the amount of trauma he has experienced, and he's nearly lost his mind.

But when you are revealed as the Listener, what he has been waiting for, what he has been holding out hope for, happens. Personally, I don't think he attacks Astrid because she isn't following the old ways. It's more like he is confused by that, and wants her to change. I think the actual reason he attacks her is because she is plotting against him and the Listener.

Looking at the tenets, I think it's also important to note that even though he attacked people, he didn't kill anyone. Even though he makes a comment about hoping that Arnbjorn bleeds to death, I don't think it was accidental that he didn't. Cicero is still trying to follow the old ways, even though he has a lot of good reasons, probably way more than Astrid, to have given up on them.