gmwylie - how many shortcuts you can take is really dependent on what attributes need to be tracked, and what the intended purpose of the weld map is.
If I have an item on a drawing with a few welds that comprise a given component (for example, a pipe support), I may be able to lump all of those fillet welds together as weld XYZ, record my welder stencils/weld material ID, and have a single inspection report for all the welds associated with that support. I may do something similar to a pipe spool, but number each individual weld joint. Likewise for a widget, etc.
However, lumping welds together isn't always ideal, especially if I'm relying on that weld map for progressing/statusing completion (i.e., I have no idea how much of my widget is really complete, because my inspection now encompasses ALL welds on that item, and my weld map just shows it as being incomplete.) However, if all you care about it being is a permanent record, then you can arguably take a drawing, draw an arrow on it, and record your pertinent information without even assigning a weld number.
Hard stamping/scribing also has its merits as racookepe1978 has mentioned, though you can run into problems depending on scale of production, especially when you realize you've missed an inspection and the part is covered in dirt/insulation/refractory linings.