Fusion—formerly known as Fusion 360®—is a cloud-based parametric CAD application like Inventor® and SolidWorks. It offers a range of tools for 3D modeling, simulation, and prototyping, as well as advanced capabilities like generative design and CAM. For additional functionality, add-ons or extensions are available for purchase. A few of these extensions are Machining (CAM), Generative Design, Simulation, and the Fusion Manage Extension.
Fusion Manage Extension, formerly known as Fusion 360 Manage Extension, is an extension for Fusion (the cad application) that can be used to participate in Fusion Manage (the PLM application) workflows like ECOs and release management from within Fusion. (There are other Fusion Extensions like Manufacturing, Simulation, Signal Integrity, and Design.)
Fusion Manage—formerly known as Fusion 360 Manage and Fusion 360 Manage with Upchain. This Fusion Manage "family" consists of 3 versions, listed from highest to lowest: Fusion Manage Enterprise, Fusion Manage, and Fusion Manage Participant. The rebranding can be confusing because when someone says "Fusion Manage," they could be referring to the entire family or the middle version of the PLM tool. I've listed the features of each of these versions below.