Both are correct. One reflects our approach before conversion and the other is true for after conversion to Judaism. Let me explain in brief.
In general, we do not encourage someone to convert. There are two reasons for this:
First of all, we believe that when a gentile keeps the seven Noachide laws, he merits a portion in the World-to- Come, and therefore there is no imperative for him to become Jewish. If, like the Christians and Moslems, we believed that those of other religions are condemned to damnation, then we also would desire to convert people. However, we believe that a person can be completely righteous and merit the World-to-Come without conversion, by adhering to the basic moral laws revealed to Noach. Therefore we feel no compulsion to convert others, unless they show a desire to convert.
Secondly, since sincerity is one of the criteria for conversion, one way in which we can determine that the candidate is sincere is by discouraging him from converting. If he persists and does so for the love of Judaism, then we accept him with open arms.
It's interesting to note that some of our most famous scholars were converts, or descendants of converts. Rabbi Akiva was the son of Yosef the ger tzedek, the righteous convert. The Talmud states that some of the greatest rabbis were descendants of Haman! The standard Aramaic translation of the Torah that is printed in most Chumashim was written by a convert, Onkelos.
However, if someone approaches a Jewish religious court (Bet Din) and the judges of the court determines that he sincerely wants to convert to Judaism, they accept him and supervise his process of conversion according to Jewish law.
Once a person sincerely converts to Judaism, they are 100% Jewish, and we are obligated to love, welcome and accept them into our people. When they recite the prayers, they refer to the Patriarchs and Matriarchs as "our fathers" and "our mothers." At the Passover Seder they say "God took our fathers out of Egypt." They are full-fledged members of the Community of Israel.
