The Question: "A new project manager joins a project during the delivery of a critical milestone. As this has multiple stakeholders, the project manager needs to determine which stakeholders should receive information about the delivery."
The project's stakeholder register should be the complete list of stakeholders. If you look at the many of the processes, one of many of their outputs is often the stakeholder register. During the life of a project we are constantly updating, discovering and refining the register. This is an iterative process which may be initiated through other processes or simply done independently as new stakeholders are discovered (i.e. the company just hired a new VP of something and he/she is now a stakeholder). (PMBOK, pg. 505, emerging trends).
There is no questions that the stakeholder engagement plan and the communication plan are linked, but the PMBOK separates these two processes. You will notice that the stakeholder engagement plan is an INPUT to the communications plan, but not the other way around.
From PMBOK (pg. 522) The stakeholder engagement plan is a component of the project management plan that identifies the strategies and actions required to promote productive involvement of stakeholders in decision making and execution.
PMBOK (pg. 377) - The communications management plan is a component of the project management plan that describes how project communications will be planned, structured, implemented, and monitored for effectiveness.
From a high-level, the stakeholder engagement plan is about "engagement", it is after all THEIR project. The communications plan is how, when, who gets project communications. Releasing project communications can also be sensitive and so the communications plan includes releasing confidential information.
Now to the question. We are wanting to know who should get this project update that we are nearing a critical milestone (which is probably a report). This is clearly in the realm of the communications plan. That being said, how do we actually know who to send the report to? Our communications plan is probably role based and our risk register can be tailored to include project roles or information level (sensitive, weekly status, milestones, financial, vendor, end-user, etc). Thus, any "financial" reports would go to those stakeholders that needed that information (project sponsor, business office, PMO, etc).
Hope this helps.