This is indeed an interesting thread, which I will join at this late stage to share my experiences.
You ask what is an electrician that can do more than just a bit of wiring. Simple, that's normally known here in the UK as a "Technician" a Technician is a halfway house between a "wireman" and an "Engineer". All three are supposed to describe specific jobs which require specific skills and qualifications.
a Wireman is usually someone with some basic training, perhaps on the job training or even an apprenticeship. He usually just builds machines, or wires plant and machinery to specific instructions, i.e working from a set of drawings that someone else has prepared. He doesn't necessarilly need to understand how the machine works.
A technician is normally apprentice trained plus some other qualifications, such as a HND here in the UK. His role is more involved. He is expected to know how machines work, understand PLC's and program them, fault find and fix them, understand drives, control logic etc etc. A more technically challenging job, but probably not doing very much real wiring.
An Engineer usually has at least a degree as his qualifications, but usually has less practical training. He's the guy that sits in a design office and prepares the schematics and panel layouts for your machines. He's expected to know a lot of technical stuff, including all sorts of programming. He rarely if ever gets his hands dirty, and probably wouldn't know what to do with a crimp tool if you gave him one.
Now I have my own chip on my shoulder, because I have met a LOT of "Engineers" who have good degree's, but quite frankly have virtually no practical knowledge and would be dangerous if they were let loose to fault find or repair a machine on their own. I have lost count of how many times an Engineer has asked me to fit a mains plug onto a bit of flex, because they don't know how.
It sounds to me like the original poster of this thread is looking for a technicians job. Well here in the UK you would find that hard to get without experience, so I guess what you really want is someone who will take you on as a trainee technician, and give you the necessary training, and oportunity to gain the necessary qualifications.
You might be wondering what job I do. Well i'm apprentice trained, HND qualified, and have 25 years experience working on industrial control equipment. I now do mainly machine design and PLC programming, but also get involved in commissioning and fault finding. I do little wiring now, but in the past I was just a wireman. I'm actually doing a job that when advertised was as an "Engineer" and required a degree qualification, and my sallary is that of an engineer. Some people actually call me an engineer. But hold on, I don't have a degree, so i'm not "officially" an engineer, but a technician. But I do seem to have a rather unique (among engineers) ability to actually understand and appreciate how to wire a machine and how to fault find a machine. Something which a lot of my more highly regarded "propper" engineering colleagues (the ones with degrees) don't have. Hence my chip on my shoulder, because I have the ability to do the job of an engineer and a technician, but a LOT of employers won't even consider me for an engineers job, because I don't have a degree.
If asked to reccomend a carreer path, I would undoubtably say the way I did it was best. Start with an apprenticeship and work your way up with experience and on the job training.
