Construction Management

I have a fairly solid understanding of the role of a CM in project delivery. I am a little confused as to the variety of terms used and if there are overlapping descriptions.

CM as Advisor

CM as Agent

CM as Constructor

CM at risk 

There seems to be a distinction between the Advisor and Agent role where the Advisor is strictly a consulting presence who conducts constructability and cost analysis for the project, while an Agent will do those things as well as have a more hands on role in selecting and/or coordinating design trades throughout the construction process. Both of these roles are without risk to the CM to construct the project on time and within the established budget.

I have at times seen the 2 roles used interchangeably and I'm curious if anyone knows whether ARE exams distinguish between the 2 or see both as generally similar roles in project delivery?

The CM as Constructor will perform constructability and cost analysis and then at some point switch over to the traditional contractor/builder role and be responsible for adhering to the schedule and budget that they themselves have worked to generate. 

My question here is that I have seen the term CM at risk used independently of CM as Constructor and I am wondering if it is the same exact role with different titles (based on different info sources) or if there is a nuanced difference I should be aware of? I have only ever seen CM at risk described in generally the same terms as CM as Constructor.

Thanks