According to the name index in Strasburg Lutheran German Records, 1768-1829, Haack, Haag, and Hag are variant spellings of the same name. I think that the family probably originated in The Hague, Netherlands. I used the spelling Haack because of this entry in the baptismal records on page 18 [page 17 in the German original]: "(Parents) Johannes Krodhauss & Elizabeth; (Child) Anna Maria; (Birth) 17 Nov 75; (Baptism) 15 Sep 76; (Godparents) Joh. Krodhaus & Catharina Haack (single)."
In the list of those who took Communion on April 9, 1775, her name was spelled Catharina Hag.
In the baptismal entries for 15 Nov 78, John Greathouse & Catharina Elissen were listed as parents of the child Elizabeth, and the godparents were Tobias Haag & wife Catharina Dorothea. [The next entry is the baptism of Johannes (John), son of John Greathouse & Elizabeth.]
Since the pattern of listing names throughout this record is to list the surname of the woman only when she is single and to list her first or first and middle names when she is listed with her husband (whose full name is given), it is obvious that Elissen [Elizabeth or Eliza] was Catharina's middle name.
I would be in favor of using the Anglicized form of her name--Catherine Elizabeth Hague--if the German form is a problem.
Vivian Morlock Taylor