By the way Morrisons news article is incorrect they do NOT run dbs checks on staff only the basic standard or basic Scotland back ground check as it is for all supermarkets I believe.
A list would be quite hard to do or manage but yeah would solve many issues if people knew which employer runs which type of check this way we would not waste hours or time getting hopes up etc.
But the general jist of it is if its a job role working directly with children or elderly then a dbs check is done which shows ALL.
Financial services -ie banks or where large amounts of money, responsibility etc.
Offices, Warehouses, retail, shops, supermarkets even charities on the high street they should only be allowed to run basic Scotland checks. Also to note the "google effect" make sure hopefully your past is not advertised on newspapers or internet articles, many companies do a google check with your name to see any dirt to misuse against you.
In regards to your question above not sure actually, sounds like they may ask for a basic scotland check or they may ask for a dbs check- again if you think the job position is not any on the small list I said above you can contact dbs and dispute it or not fill that form in tell your boss to be you need to check with dbs offices if its ok, if they find that funny perhaps suggest you had an issue with dbs last time.... remember the dbs check comes back to your home address only.
What job role type is it anyhow?