I agree that Philip's was a royal coronet. Did not use my notes.
Here is one note I found - In the book for the coronation of Edward VII, and dealing with the regalia --
" The coronet of the Prince of Wales differs from that of a duke in having alternate crosses-patées, and fleur-de-lis of gold instead of strawberry leaves, with a single arch depressed in the centre and bearing an orb or mound surmounted by a cross. " The coronets of the princes of the royal family are the same, but without the arch, and those of the princesses substitute strawberry leaves for two of the crosses. " It is said that Charles II prescribed all these differences."