So you have a data column named My Color Names? You have levels in that data column like Aqua and Black? If so, JMP does not interpret those names and produce the named color. Instead, JMP interprets the names as categories or groups and it selects a default color for each level in order. The value ordering in this case is alphabetic.
You could assign the desired color as a row state in the data table. Select each group of rows and then select Rows > Color and then the color that you want to use. Another way to use the color row state is to select Rows > Color or Mark by Column. Select the My Color Names column. You will now see the same color scheme that Graph Builder by default. You can change the color theme as a whole or edit the individual colors.
The other way is to edit the colors in the Graph Builder legend. You can change the overall color theme or individual colors.