You won't find one single neat definition that always applies, because the terms are used differently for different materials.
For example with metal stock, a bar can be both round, square or rectangular, but has only rough tolerances. Rod had been precision ground to within very narrow tolerances, and is circular in cross section.
With glass, we generally only refer to rods.
We use the term 'curtain rod' to describe hangers that vary widely in terms of size and diameter, yet they're all lumped under 'rod'. However, they all do conform to a relatively limited number of standardized lengths.
Arc welding uses a metal wire coated in flux most of which are around 1/8" diameter, give or take a few sixteenths. You see the terms both 'stick' and 'rod' used for those expendable electrodes.
With wood, we generally only use 'stick', although the thick wooden beams used to block doors are sometimes called 'bars', hence the term 'bar the door'. 'Dowels' are round wood shafts of different diameters and lengths. With wood, 'stick' also signifies it's naturally occurring, but such is not the case with a stick of welding wire.
I don't recommend you bother yourself too much with these definitions, because they vary so much and aren't always consistent. Also, it's difficult to imagine a situation where accidentally using the 'wrong' term is going to impede comprehension. You might get a strange look if you start talking about a 'curtain stick', but context is going to rapidly clarify what you mean, and at worse someone might correct you.
So many of these usages are more a matter of learning common collocations, rather than memorizing strict definitions.