Is a training record the same thing as evidence of competence?

It is not, and the difference surfaces in an audit. A training record shows that someone was present; evidence of competence shows that the person can do the work and has accepted that they can. The second needs a signature, because a commitment is something a person takes on, not something explained to them. Commitments also expire: one signed a year ago and never renewed still exists as paper, but no longer stands as evidence. SenSight QMS says this plainly: evidence is a signed commitment, and an expired one is not evidence.
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In SenSight QMS: the personnel screen raises expired commitments as a separate warning.
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SenSight QMS — quality management and calibration records in one place

SenSight QMS keeps the record side of ISO 9001 and ISO 17025 in one place - for accredited calibration and testing laboratories, and for manufacturers who run quality separately from their ERP.

SenSight QMS