Does anyone have to hand the reference that defines a failed LPC or LST as becoming PUT?
You are asking the question the wrong way round:
The Examiner will always be Captain and so will Log PIC in all cases.
Therefore, in Single Pilot Aeroplanes, the Candidate can only be PUT unless there is a rule somewhere that stipulates otherwise - and in the UK there is, but this applies only to successful tests.
So, as Whopity quoted, there is a reference that defines a passed LPC or LST as becoming PICUS.