The rite of circumcision is found in Gen. 17:9-14, 23-27 when Abraham
became party to the covenant.  Ishmael who was 13 at the time of the
commandment was circumcised at that time.  Isaac was born later and
circumcised on the 8th day after  his birth which is why we Jews perform
the circumcision on that day in a Brit Milah.  This is also the time when a
Jewish male receives his Hebrew name.  I am not an authority, but I don't
recall anything with regard to Jewish custom being other than full
circumcision of the foreskin.  Actually, I think circumcision pretty much
means "cut around" if translated from the Latin.