Many of us use this term in daily business quite intuitively: "build stage", "test stage" and used long before the rise of DevOps.
But is there a formal and/or well-established definition of this term?
Additional question: Who and where used it for first time?
Often in technology we co-opt English words and give them alternate meanings. However, in this case it's really just the standard definition:
A level, degree, or period of time in the course of a process
I would argue that referring to these milestones as "phases" would be preferable to "stages." This is because "staging" is already a more prevalent term in use in the industry.
Staging refers to a specific environment used for walking through testing the deployment process. It is distinct from the QA environment used for bug testing and sanity checking and testing environments used for User Acceptance & integration Testing (UAT). All of these could be considered types of test environments.
The origin of this term comes from theater in which actors will walk through a scene - often stopping to adjust, clarify, and note physical positions on stage. This is often also referred to as "blocking" in theater.
Thus, if you call these milestones "stages" you could end up having a "staging stage" and for the sake of clarity, "staging phase" or "staging milestone" would be preferable and more prevalent. These are the more common and customary terms, driven by the terminology of project management.
I would typically dis-recommend using the term "testing" to refer to an environment, because it can be confusing as to which testing environment you are referring.
stage term, the staging term is indeed your definition and I didn't say anything else. It's just a counterpoint to your first sentence which is false. A staging phase would be running the deploy stage toward the staging environment. Here phases are build of stages. Stages are construct the same way in each phase and differ from their input only. See the phases as a rocket launch, the staging env as the rocket destination and the deploy stage as one stage of the rocket, and you have a common picture of a deploy pipeline nowaday.