The idea that there is a 'King's Crown' and a 'Queen's Crown' is false. They are, respectively, the 'Tudor Crown' and the 'St Edward's Crown'.
It is up to the Monarch to choose what he or she wants symbolically.
Exactly so. However, as the last Queen Regnant (Victoria) chose St Edward's and all subsequent Kings chose Tudor it is not surprising that they are mis-called sometimes; assisted by their use on the VC and the GC.
What is more worrying is that the chart at Post 4 above calls it the "QE II Cypher". No it ain't; the cypher is the "postman's badge" of E II R as any ful kno, not the Crown.