It appears that you are describing the staged representation of CMMI v1.2 or v1.3. These versions of CMMI also have a continuous representation. CMMI v2 has a different representation of maturity models that has some aspects of both the staged and continuous representations.
Since it's the latest (and last) version released by the Software Engineering Institute with open information, I'll discuss this question in the context of CMMI v1.3. Much will hold true for CMMI v1.2 as well, but won't be as correct in the context of CMMI v2.
The names "Managed", "Defined", "Quantitatively Managed", and "Optimizing" are the names that the authors of the CMMI chose to represent their levels. They do invoke some meaning behind the intent. An organization at level 2 is capable of managing their projects and processes, but they don't have a full set of defined processes across the key areas. The later levels focus on using quantitative measures of their process and ways to optimize their processes.
If an organization has achieved a maturity level of Managed, it means that they have successfully documented and can demonstrate a number of processes. There are multiple flavors of CMMI (Acquisition, Development, and Services). For a Development organization, it means defining processes in Project Monitoring and Control, Project Planning, Requirements Management, Configuration Management, Measurement and Analysis, Process and Product Quality Assurance, and Supplier Agreement Management. If they have been appraised at this level, an appraiser has reviewed their documented processes and what the organization actually does and there is alignment.
The staged representation of CMMI looks at these processes as a set. There is a set of processes that must be defined and assessed against for a Managed organization. A Level 3, or Defined, organization, must meet all of the criteria for a Level 2 organization plus implement additional process areas. The same is true for Levels 4 and 5. However, the continuous representation looks at the maturity of individual process areas, allowing the organization to one or more process areas to be evaluated against their own goals and objectives.
When CMMI was more popular, CMMI Level 3, or Defined, was often a target. At one point in time, being appraised at CMMI Level 3 was a consideration for receiving US federal government contracts of a certain size. Some studies have also indicated that the biggest "bang for the buck" in terms of balancing process control with real-world outcomes came at CMMI Level 3, and the cost of fully implementing the controls at Level 4 and Level 5 was burdensome.