The inspection is limited to the weld attributes listed by the applicable code.
For instance: ASME Section VIII does not include criteria for undercut. Therefore, the inspector cannot evaluate undercut and the welds cannot be rejected for undercut. The exception would be for those instances where the manufacture's quality program included criteria for undercut. Remember, the manufacture's quality program can be more stringent than the code, but it cannot be less stringent than the applicable code.
Back to the undercut; while the inspection criteria of Section VIII does not address undercut specifically, it does have a few words and criteria for "base metal thinning due to the manufacturing process." One would use that criteria for evaluating the undercut without using the word "undercut" in the report.
As for your situation, there is no criteria for the width of the cover layer if the governing standard is AWS D1.1, affectionately referred to as the "New Farm Code." If the width of the cover layer is a problem, the contractor can address it in the quality program or the Owner can address it in the project specifications. The visual inspector must evaluate the welds based on the criteria provided. It is not the inspector's prerogative to "make up" the inspection criteria based on personal preference or based on personal opinion.
Best regards - Al