In my opinion, DevOps is first and foremost a topic of organizational culture.
In contrast to my story, developers and operations often have conflicting interests. Developers prefer to have an easy, fast, frequent deployment process. Operations prefer a stable, high quality release, so they are often more conservative.
So the goal of DevOps would be: bring these two viewpoints together by collaborating closely. Have easy, fast, frequent AND stable and high-quality releases.
On the operations side, that would mean: gain empathy for the development side. Learn about developing automation of operation functions, like Infrastructure as Code.
On the development side, it would mean: gain empathy for the operations side. Use a continuous delivery pipeline with different kinds of automated tests. Lean about Monitoring, Rollbacks in production, Chaos Engineering and so forth.
I think both of the scenarios you describe are feasible: having all team members with DevOps skills, or having a separate team.
But if you have a separate team, this could be a smell that the cultural prerequisites are not there yet.
So the question would be: do the developers and that operations team work towards common goals?
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