The DVSA's official advice is given in their publication "Driving - The Essential Skills", which is part of the curriculum for the driving tests, and which all new drivers should have read. Since at least the 2005 edition, it has recognised that there are two methods, and says very clearly "Refer to your manufacturer's handbook to check the correct procedure for your vehicle".
The handbooks for every car I had between about 1995 and 2015 told you NOT to press the button. (None of my more recent cars have had handbrakes).
I wouldn't claim superior knowledge or experience, but I suspect the latter may be more recent than that of your contacts. When I was instructing I never taught pupils to press the button, and it never was an issue in any of 100s of tests (many of which I accompanied), nor raised by any of the examiners with whom we had regular discussions.
If your daughter really failed her test for this non-fault, then the examiner was wrong to mark a fault, and doubly wrong to categorise it as serious or dangerous. She had valid grounds for complaint.
FWIW I understand that the manufacturers' rationale for the changed advice was that the consequences of accidental non-engagement with the button pressed were infinitely greater than those of possible eventual wear on the mechanism.