firstpass,
As far as I can recall, or find, D1.1 itself does not have a 'requirement' for tracking welder to weld. But D1.8 does as a supplement to D1.1 when required for Seismic Designs. It is 6.13 of D1.8. It calls for marking of all welds with welder's ID mark. From there it is a matter of logging the info into a spreadsheet and all depends upon how detailed of a log you need for tracing the information. Some jobs require a very complicated weld track system.
I don't work in B 31.3 so can't answer that one for you.
Hope that helps.
Have a Great Day, Brent