If you find Friederich's name spelled more than one way on this site, that's partly because Friederich himself spelled it different ways. When he and his family came to America in 1829, Germans were still a small minority. Eager to assimilate, he used "Frederick Smith" more often than his German name.
I normally consider the gravestone authoritative, so for years I spelled his name "Friedrich." More recently, however, I found his signature on his will. Written in the German Kurrent he learned as a child, it clearly says "Friederich." But after years of spelling the name differently, I sometimes slip up and revert to the old way.