With time, the surname used might be different from the "legitimate" birth registration. So German terms came to be used to describe the legal surname vs. the surname used.
"False" (often abbreviate f.) implies usage and means an individual might be using the father's surname but officially should be using the mother's surname due to lack of the required civil marriage. Example: Brown f. Cohen means the person used Cohen (father's surname) but Brown is the legal name (mother's surname).
"Recte" (often abbreviated r.) means "legally" or "correctly" or "corrected" and was used to correct the surname of an individual, sometimes following a civil marriage between the parents. Example: Brown r. Cohen means the person used Brown, but Cohen is the legal name.
"Vel" translates to "also known as." The resident was known by both surnames, maternal and paternal. If vel appears between two given names, this might signify the Jewish versus secular surname or nickname/kinnui, versus legal given name. Vel was used mostly with given names and false/recte with surnames.
Phyllis Kramer
Palm Beach Gardens, Fla
Director, Education, JewishGen Inc: https://www.JewishGen.org/education