There is a post with a good method of plug welding. weldeng13 give a good reference for making a plug weld. I do not believe that a plug falls into the groove weld or fillet weld category, It can be either a complete or partial penetration weld.
troilite, AWS D1.1 2006
4.29 Plug Weld and Slot Welds for Tubular and Nontubular Connections
Qualification for CJP groove welds ...shall qualify for all plug and slot welds.
Table 4.10 note b Groove weld qualifications shall also qualify plug and slot welds for the test positions indicated.
I don't think a fillet weld procedure or symbol is appropriate.
Table 4.11 Plug (Fig. 38) ( 3/8" test plate) Qualified Dimensions (Nominal Plate, Pipe or Tube Thickness Qualified)= Unlimited thickness
For Groove weld operator qualification it is 2T of test plate.
Table 4.2 WPS Qualification-CJP Groove Welds- Test on plate 3/8"<T<1" is 2T max thickness.
My answer is 1.) plug welds are considered CJP, 2.)Your minimum WPS plate thickness should be 1/2" thk. 3.) The symbol should be plug weld not fillet weld.
You mention ASME IX, aren't you using D1.1?