The authority of the CWI is limited by the authority delegated by the Owner or the Owner's Engineer. AWS B5.1, Table 1, lists the job responsibilities a WI (Welding Inspector/CWI) or SWI (Senior WI or SCWI) should be capable of performing. However, as the third party inspector and if the governing document is AWS D1.1, the duties of the Verification Inspector (TPI) are quite limited.
There are many instances where the Engineer (again, per D1.1) can and does delegate certain responsibilities to the CWI/SCWI.
As for the CSWIP, not a clue.
Best regards - Al