Who decides you are Jewish – a neighbor, a rabbi or you yourself?

With all non-Jewish faiths and ethnicities, their definition is based on how somebody defines him/her-self. If somebody says, I am a Catholic, or an Assyrian, or a Russian, or an atheist … the others accept it without any doubt. With the Jews, everything is different.
From the news media:
The boxing website thesweetscience.com reported on Wednesday that Muhammad Ali attended Shabbat services at Congregation Rodeph Shalom in Philadelphia, Pennsylvania, in April, to celebrate the bar mitzva of his grandson, Jacob Wertheimer.
Ali’s daughter, Khaliah Ali-Wertheimer, was raised as a Muslim, but described herself as “not into organized religion” and “more spiritual than religious.” Her husband, attorney Spencer Wertheimer, is Jewish and their son, according to Khaliah, chose to have a bar mitzva because he “felt a kinship with Judaism and Jewish culture,” the Inquirer reported.
According to Islam, a child’s religion is passed down from the father, while according to traditional Judaism it is passed down through the mother. This means that Islam, the religion of Jacob’s mother, considers him to be Jewish, and his father’s Jewish religion considers him to be a Muslim.
Thus, Jacob Wertheimer himself has decided he is Jewish, a Jacob’s Jewish orthodox neighbor has no doubt Jacob is not Jewish since his mother is not Jewish, and Jacob’s non-orthodox rabbi, knowing that Jacob’s mother is not Jewish and therefore Jacob has to be considered non-Jewish, nevertheless performed a bar mitzvah. In defining the Jacob’s Jewishness, who is right and who is wrong?
From the news media:
Application that identifies (((Jews))) online disappears from Google Browser. The developer of the Google Chrome extension Coincidence Detector, which identifies Jewish individuals and organizations by putting triple parentheses around their names, pulled the application last night after several news reports exposed it to the general public.
In this case, Google developers were deciding who is Jewish and who is not.
From the Hitler and Stalin’s directives:
A person with 1/8 of “Jewish blood” should be considered Jewish, or a person, which is identified as Jewish in his/her passport, is Jewish.
Here the government decides who should be considered Jewish.
It looks like almost everybody in this world has an opinion on who is Jewish and who is not. If it is so, what might be a right way to define the true Jewishness?
The true Jews are those who follow in their individual lives the original spiritual ideas precisely described by the Chabad in the following way:
Throughout our 3300-year history, what has defined us as Jews is a relationship and commitment. We are Jews because G-d chose us to be His "cherished treasure from all the nations... a kingdom of priests and a holy people" (Exodus 19:5-6). We are Jews because G-d chose us to play the central role in the implementation of His purpose in creation: to orientate our lives in accordance with His will, and to develop a society and world community that reflects His goodness and perfection. The substance of this relationship, the charter of this commitment, is the Torah.
Thus, the true Jews is somebody who is applying the Torah moral principles to his work on developing a society and world community that reflects the Torah guidance. Only the true Jew, who may call him/her-self religious or non-religious, may decide for him/her-self how to apply the Torah guidance to all that. If he/she feel uncertain how to apply, a rabbi may be asked for an advice – but for an advice, not for marching orders.
If it is so, an individual have to decide for him/her-self on being Jewish – not a neighbor or a rabbi.