Let me explain something simple to you since clearly your understand of this is limited to a few readings in between. If you convert to Judaism (Reconstruction, Conservative, or if it happens Orthodox)union wrote:
Who wants to convert into the Haredi? It's people who actually are converting out of religious conviction, since Haredi are like the only Jews who follow the 613 commandments of the halakha, or Jewish law. Most converts to reform and conservative Judaism, like Giffords, convert either for marriage or image reasons. That's why the Orthodox call such conversions "fictitious and insincere", which is often the case. Also, you're wrong in the sense that Orthodox Jews accept reform Jews as being actually Jewish. As long as the reform jew doesn't have a gentile mother, he/she will be considered Jewish. If you have a Jewish mother, you will be considered universally Jewish by allt he sects, even if you are an atheist or whatever. The reason some American Jews face a lot of doubt is because of their high intermarriage with gentiles. 50% of American Jews marry a gentile.
If your mother is a Jew, you are half-Jewish by ethnicity, and fully Jewish according to Jewish law.
If your father is a Jew, you are half-Jewish by ethnicity, and not Jewish at all according to Jewish law.
Atheists who are Jewish by ethnicity, may consider themselves Jewish by ethnicity only (and allowed to be stoned according to the Torah) .
Gabrille Gifford's father is Jewish and her grandfather an Orthodox rabbi. According to the Orthodox, since she converted to Reform (not considered Jewish by the Orthodox don't know where you got your info) and her mother was not one she is not Jewish by religion or ethnicity.
Being fully Jewish is a cross-section of being religious Jewish and a Jewish ethnic identity built over thousands of years in exile and a specific insular sub-group.
One of my best friends since 6 grade is a Conservative Jew and I witnessed both bat Mitzvahs and some holiday meals including my extensive reading on them. Don't argue if you don't know.

