What you heard from your cousins is, in fact, the truth. Whether one chooses to become Jewish as an adult, or is adopted as a child, they don’t technically become Jewish until they convert in accordance with Jewish law.
To explain: the Jewish people is not a “race” – there are Jews of every race. We also do not share a common country or location, (most Jews have not lived in Israel for nearly 2000 years). We are not a nation based upon language, as most Jews don’t speak Hebrew. Although we have a common religion, a Jew is still fully Jewish even if he is not observant. If so, what makes one Jewish?
The answer is, what makes someone Jewish is a “Jewish soul”. The entire Jewish people were endowed with a unique soul at Sinai. The Jews received souls which would be capable of receiving the powerful spiritual energy transmitted by the Almighty through the Torah. These unique souls are instilled in Jews throughout the generations, as the Jews continue to be the torch-bearers of the Torah and all the spiritual light contained within it. This is our mission as a “light unto the nations”. For this reason, the Midrash says all the future Jewish souls participated in the transmission of Torah at Sinai, so they would be fit to be endowed to a Jew who can study Torah and be a light unto the nations.
This type of unique soul is endowed to a convert to Judaism as well. He or she receives one of the souls which were present at Sinai. In this way they become part of the Jewish people. The conversion process was first carried out at Sinai, when the entire nation received expanded souls. The convert continues that tradition and – like the Jews of Sinai – joins the mission of the Jewish people. When a convert agrees to accept all the mitzvos and live a Jewish life, he or she are having their own private Sinai experience and are endowed with a unique soul.
This process works for a small child as well. It is true that the child does not yet have the wisdom to make that decision, but the Jewish Court (Beis Din) are empowered by the Torah to make that decision for the child, as it is considered a merit for the child. With the power of Beis Din they become Jewish and are endowed with a Jewish soul.
The child who was converted does, however, have the right to reject their Judaism if they so choose. For this reason we ask a boy at age 13 or a girl at age 12 if they are happy with becoming Jewish. If they do not reject their Judaism then they are fully Jewish forever, like any other Jew.
I wish you the best of success in bringing this process to fruition and entering your children into the eternal covenant of Abraham and the Jewish people, with much nachas!
Sincerely,
Rabbi Yerachmiel Fried