English and German form of Beatrice, from Latin "bringer of joy" or possibly a variation on "pilgrim".
A number of alternative spellings of Beatrice and Beatrix are found in the 1841 Census of Aberdeenshire. Variants of Beatrix are listed above. All should be found by using a Soundex search.
There is a further group of spelling variations of Bettridge [listed under that name] which are also variants of Beatrice/Beatrix.
According to FreeREG, Beatrix was used in old documents for Betteris, while Beatrice is a 19th century name in English-speaking countries.
ONC says the Scottish Gaelic name Beathag, a variant of Beatrice/Beatrix, is "often Anglicized as Sophia or Rebecca" so, in Gaelic areas of Scotland, these four names can become lesser synonyms for each other.
While there appears to be no obvious etymological connection between Beatrice/Beatrix and Elizabeth/Elisabeth, anecdotal evidence [AC, BA, BH/GMcC] exists for the names being interchangeable (i.e. synonymous) in Scotland, possibly through the common pet name, Betty. One correspondent [AK] suggests that the names are etymologically similar but hard evidence does not appear in official sources. In "Tracing Scottish Ancestors", Rosemary Bigwood asserts that Beatrix is Anglicized as Elizabeth but supporting sources are lacking.