+1 for the aged Morcambe & Wise reference
When we talk about "full stack" software, we are most commonly talking about the disciplines and elements (and their associated competencies) that span the complete smokestack of software application development as we typically define it. Laurence Gellert's well-composed piece in 2012 nicely clarified this stack as the strata …
I'm DevOps'd to the max so I know what I'm talking about here. As long as you are automating "all the things" then you are doing it right. It says so on my LinkedIN profile, and recruiters say I can charge a premium for it, so it must be so.
If you aren't sure you are doing it right, I can come and speak at your conference. For a hefty fee of course.
Oh, I also qualify as the "Full Stack Exchange" developer you mention at the start.
One problem is that DevOps is becoming a bit like photography or web-site creation - just about everybody thinks they know how to do it, but most people's attempts are mediocre at best, and even fewer understand what it takes to do a good, consistent, professional job.
In essence, DevOps has been around for years, but it's only recently that it has been given a name and more formal definition. That has sparked a host of ideas and approaches, many of which conflict. At the same time, there's a vast slew of new tools available, each claiming to make life easier; but mixing the wrong set of tools can quickly create a system of unbelievable complexity, or force teams to work in counter-productive ways.
DevOps probably should be a stack, but in practice it's often a pile - that pile too often being a mixture of legacy systems, new and untested DevOps tools, hastily implemented automated testing and maybe a kludged attempt at continuous delivery.
If management isn't convinced of the need for proper investment in DevOps then it's a potential recipe for disaster. Becoming an Andrew Preview is only really possible if you have the full support of the management comedians around you ...