Interesting difference you want to differentiate. I think you almost get there. The main point in difference is that what you are going to manage ie what is your objective of the management area. If you would like to manage Project Information, then you are managing "Manage Communications" (according to set communication plan using modes, methods, technologies etc of communications). But if your purpose is to manage stakeholders engagement then you are following process Manage Stakeholder Engagement (meaning gaining their support, reducing their resistance, make aware who is still aware etc). Actually you are managing needs and expectations of the stakeholders by employing Manage Stakeholder Engagement process BY USING MANAGE COMMUNICATIONS. Hence, in this regard Manage Communications is a MEANS to achieve the purpose of Manage Stakeholder Engagement. Thanks for your thought.
Mohammad Sikder