What positions in industry should I apply for after a phd and postdoc in software engineering? This could also be not the most appropriate place where to post, but I'll try anyway.

Some background info: I am about to finish a 2-years postdoc in software engineering, and I am evaluating a number of options, including moving to industry. I graduated at 22 (bachelors), 24 (masters), 27 (phd), and now I am 29. In summary, I have a quite strong academic record (top grades from respectable universities, ~20 publications in acknowledged conferences and journals), but never had a position in industry. However, the type of research I conducted in my phd and postdoc has been in close collaboration with industrial partners on concrete software engineering challenges, so I'd say I have way more than just an idea on how the software process works in industry. Over the years, I also had the opportunity to grapple with management-like tasks in research: write reports/proposals (including some form budgeting), coordinate the delivering process of (guess what?) deliverables, co-supervise and advise students, peer-review.

As for what I would like to do, I see myself as performing best working as a software process improver. Given my background, this means design solutions for (ideally) smarter testing, reducing effort in the maintenance of software products. Of course I can also implement these solutions and know-how in a kind of "internal" tools (when needed), but the biggest contribution I would give is an answer to "what should we do to reduce costs in testing", rather than implementing some code that does it. I envision that I can relatively easily do so in other phases in the software lifecycle (for sure requirements/change management, where I also have a kind of background, maybe a bit less analysis and design).

I am a bit unsure on what kind of positions I should look for with my background. On one hand, I never ever have been keen on programming/developing/coding. Of course I am perfectly able to do this, but I have to admit I choose to study for 10 years to develop critical thinking skills and high-level principles, rather than focusing on the flavor-of-the-year technology.

On the other hand, most job postings for things different than "write code" (in any of its sugar-coated variations such as "work under the hood", "get the job done and not just talk about it", "be a doer", "power-drive through technology", etc) are so cryptic that the few ones you actually understand are those for which you know you are not relevant. Expert in TV decoders firmware and signals? Good! I am not your guy.

On the third tentacle, "higher level" (senior or not entry-level) job postings require 3/5+ years of experience in "insert something non-academic related here". Well, I have 3/5+ years of experience with "working in collaboration with industry", which can be easily extended to include whatever buzzwords are fashionable this days: DevOps? Sure, everybody is agile today. IT Operations? Well, I didn't do research locked in a basement. Service delivery? Everything is a service, and yes, I delivered quite a bit of things. People management and decision-making skills? Of course, the experience in research I had was that for some tasks you are the principal investigator and direct others on directions to explore, gather results, and decide where to progress. You might not be the boss behind the desk barking orders at his minions, but you have to take the lead in research. I am just not sure whether all of this counts, or better, can be stated in such a form on a resume to be relevant.

Some enlightenment? Be cruel and evil, don't be afraid to say that I should "just" look at entry-level positions as if I was fresh graduate :)
We don't know if you will get one of the jobs that requires 3-5 years experience.  But there's no reason not to try.

As you're reading job descriptions, avoid the ones that clearly involving sitting and coding for eight hours a day -- because you say that's not your strong suit.

When you apply for a job you think you would be good at, you must tailor your CV, your cover letter, and your interview to the specific job.  Piece together the work you've done to make a jigsaw puzzle picture that resembles, to a reasonable extent, the picture they have in mind.  This does require some chutzpah.  That's okay.  The job hunt process is not the place for humility.

In other words, you will argue that you have 3-5 years' relevant experience (in some sort of equivalent form).

If that doesn't get anywhere -- then you can go back to the drawing board.

(By the way, you might want to get involved with a headhunter.  I myself can't advise for or against going that route.)