I am getting stuck finding the English word for "The person whose profession is to repair a car's body", for example removing the rust and painting the car.
A person whose job is to beat out the bodywork of motor vehicles.
Wikipedia suggests that the Canadian and US equivalent is an auto body mechanic.
A Google search for this type of thing also turns up lots of other titles, including:
Paint and Body Technician
Auto Body Technician
Paint Repair Technician
Automotive Body Repairer
I'm not sure you're going to get a single word/phrase that is universally accepted in the English speaking world. It would probably depend on local usage and what terminology is used for local qualifications.
In the US, according to the Dictionary of Occupational Titles, the term automobile-body repairer or automobile-body worker is used
body-line finisher; body repairer, bus; dent remover; door repairer, bus; metal bumper; metal shrinker; metal worker; touch-up finisher, metal Repairs damaged bodies and body parts of automotive vehicles, such as automobiles, buses, and light trucks according to repair manuals, using handtools and power tools: Examines damaged vehicles and estimates cost of repairs [SHOP ESTIMATOR (automotive ser.) 807.267-010]. Removes upholstery, accessories, electrical and hydraulic window-and-seat-operating equipment, and trim to gain access to vehicle body and fenders. Positions dolly block against surface of dented area and beats opposite surface to remove dents, using hammer. Fills depressions with body filler, using putty knife. Removes damaged fenders, panels, and grills, using wrenches and cutting torch, and bolts or welds replacement parts in position, using wrenches or welding equipment. Straightens bent automobile frames, using pneumatic frame straightening machine. Files, grinds, and sands repaired surfaces, using power tools and handtools. Refinishes repaired surface, using paint spray gun and sander. Aims headlights, aligns wheels, and bleeds hydraulic brake system. May paint surfaces after performing body repairs and be designated Automobile-Body Repairer, Combination (automotive ser.). May repair or replace defective mechanical parts [AUTOMOBILE MECHANIC (automotive ser.) 620.261-010].
This is somewhat different from one who focuses on the mechanical aspects of car repair.
In common usage, automobile is routinely shortened to auto, and often car, so you may see auto-body repairer or car-body repairer.
Consider, body shop mechanic.
My thoughts headed fast into a cesspool, all blaming Bob, the body shop mechanic, or myself for weeks of inconvenience, expense and frustration.
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This London website has advertisements for panel beaters, spray painters, automotive bodyworkers and body repair technicians. But the most numerous appear to be panel beaters.
A coachbuilder. In the UK, I think this word applies more for custom-built body work.
A person or company who makes and fits the bodywork of vehicles such as cars, buses and railway carriages.
Detailing describes the examples, if not the title question.