An interesting question, the son of your Edward Diamond Dyman is entered as "Diamond" but his grandson Matthew is entered as "Dymond" (http://www.wikitree.com/wiki/Dymond-59) Perhaps it depends on the origin of this family. If they weren't English, trying to understand the root name before emigration might help
This is what I understand - The name Diamond with its endless alternatives might come from almost anywhere. DNA testing has revealed at least eight unconnected root families including North American Indian (Algonquian), Irish, Jewish, English & European. The Diamond name might indicate a gem merchant, a dairyman, or perhaps simply be the English phonetic equivalent of a European name. The original Gaelic form of Dymond was 0’Diaman; descendant of Diamain. Irish variations of Diamond include Dyment, Diment, Dymott, Dimont, Deman and many more. In ancient Ireland, the Diamond family came from Derry.
My own Diamonds came from Ulster, Ireland to Galloway, Scotland.
