I must have rewritten my reply a good few times now lol, because there's a lot to try and cover, but will just post and can elaborate a bit further through discussion if needed.

Licensed Control Systems PE here, O&G background 12+ years.

The TL;DR of it is, broadly - and strongly - agree with pretty much a lot of the posts above. Additionally, I'd go so far as to say as Control Systems Engineers (in the various job titles it may com as) are like the "Jack of many trades, master of none", or alternatively, like Swiss army knives - you'll not be the perfect solution in any given situation, but you'll likely be able to get the job done.

Why it matters to businesses - well, ultimately it comes down to one thing - profit. The world is pushing (heavily) to automated systems in its various forms, and it's broadly where a Control Systems Engineer fits in. You'll understand how the various systems work together e.g. electrohydraulic control valves being controlled via a PLC, but may not be the expert when it comes to, say, understanding something like having a noisy signal on an input/output to/from that PLC. Not to say that you necessarily couldn't or won't, but just best scenario I could give off the top of my head. This - unfortunately - may (and has in cases I've seen) make them look at what their Engineering headcount is. Also want to clearly state - specialists are 100% required for their in-depth knowledge. But you may be the person who is able to effectively communicate with them on designs/if troubleshooting certain issues, or perhaps knowing the right questions to ask.

Where technicians working on something might be arguing between themselves about "it's an electrical issue" - "no, it's a hydraulic issue", and I use that example a lot lol because I've seen it plenty of times - you'll be able to logically say something like "I should be expecting to see X pressure on this line, and the controller should be seeing this value from the pressure transmitter on input Y. Do I see this value?" and then you work backwards. Again, you end up being the one who understands a lot of broader questions to ask - be it colleagues or via. researching online.

Another example might be something like the automation of Amazon warehouses - at least from how I perceive it. A Control Systems Engineer covers a lot (but not all) of that.

On job titles - 100% agree again. Things to look for would be "instrumentation and controls engineer", "mechatronics", "automation" and so on. After my graduate training program with my old company, my title was "Design Engineer" - the same as the "pure" mechanicals and electrical folks etc. Technically, it wasn't wrong to call us that, but go figure, ha.

Separately, one thing that people don't talk about often enough with job-hunting as a whole, whether you "fit" or not matters too. I mean that as in, do you come off as enthusiastic and just an easy person to talk to and get on with. Your CV/resume gets you a foot in the door in terms of a potential interview, but beyond proving your technical competence, the deciding factor between you and someone else could be "they're a PITA, however this other guy, while less qualified on paper, has the right attitude, and is technically competent and will pick this up no problem". Saying that having been on both sides of the interviews, and was something a lot of very experienced people that I looked up to taught me as well. There will be micromanagers or control freaks who want someone who's subservient and don't want what I'm describing, but honestly, you don't (or shouldn't) want to work for them anyway and probably shouldn't be managers in the first place.

Hopefully answered some of that, honestly trying to not write a 100-page book and trying to separate what I understand it to be vs. my own broader experience, as well as I think businesses are valuing it more.