Thank you for the clarification.
This is quite a discussion. The easy answer is no you don't have to record them. But the real question is what you need them for and who do you show them to.
For the first part, if you use them for a standardized process then are there any rejection limits if the product is off specifications? What happens if a worker responsible for these measurements slacks off doesn't reject some products out of specifications. To be more blunt who will be held accountable for a quality problem if there is no recording and sign off? These are just some questions you should consider if they affect you and the quality of the product and on what level, in order to decide if you need the recordins or no.
For the second part if it is something you show to a state authority inspector or to an auditor then yes you should have recordings to show them also.