As has been pointed out there is no title "King Mother." The mistake is a common one though, as people assume that the title Queen Mother means "the Queen's mother," which it doesn't. Instead it means "a former Queen consort who is a mother of a current regnant." The sex (and thus title) of the regnant doesn't play into it.
Dowager Queen vs. Queen Mother is kind of based on the situation and the preference of the individual. Most dowagers/queen mothers aren't known as anything beyond Queen X. When their husband is reigning they are The Queen, and when he dies they become Queen X, and The Queen becomes either the new monarch or his wife.
In the case of QEQM, what happened was at the time of KGVI's death there were 3 queens; Mary, QEQM, and HM. One was The Queen, of course, but she was also Elizabeth II, while the other was Queen Elizabeth, and the third was Queen Mary. To kind of help keep everything straight, QEQM formally used the Queen Mother title so that she could be distinguished from her daughter (the Queen) and her mother-in-law (the other dowager queen).