I'd say that much depends on a type of project you're going to manage. If the project is run in an agile way internally there's little more to learn. The biggest difference is the role of coach is kind of hit-and-run. Coach doesn't live long life with result of their advices. Coach doesn't fight an uphill battle to sustain changes which have just been implemented, etc. But then if the organization has reasonable approach the effort should be spread over whole team and it shouldn't be responsibility of PM only.
The bigger difference is in outside world. Even agile organizations often work for clients who aren't agile whatsoever and it's PM's role to map these two worlds back and forth. So basically you need to learn the way the client work and find a way to make them happy and at the same time build bridges to your teams approach to project. The former may mean working on project in a very formal way, preparing extensive documentations up front, analyzing whole thing at the beginning, little feedback from the client along the way, managing every change in a very formal way, etc. It may also mean difficult discussions with the client even if that's not what you'd prefer to do - if you work within budget constraints not every change should be embraced as not every client understands what it takes to eat the cake, i.e. they aren't getting it.
Then we have projects and organizations which are formal/very formal. Usually the first thing is to learn local office politics as in such organizations it is often required just to have a chance to succeed, let alone achieving a success. Then it's a bit like adopting to the way client works but on the other side. Whatever the current organizational culture is you're probably going to learn and accept it, at least to some point. If you're out of luck it won't be aligned with what you believe is right and it usually takes long, long time to change the culture even if you have much power.
To summarize that somehow I'd say that the main role of a PM is delivering a project while the main role of a coach is improving the organization. It doesn't mean one shouldn't do both but they should know where their priorities are. So if you ask about specific skills I'd go with adaptation. And if you ask about specific books I think it's more important to learn local specific than to look for generic answers in generic books.