The definition of Christianity I like the most was the one my professor gave in my religion class; that Christianity refers to the theological reflection about the person of Jesus Christ (who he was, what he did, etc, etc, slight paraphrase). Beliefs in Christianity include these next few points. One, Jesus Christ died on a cross to save us, then rose again three days later, promising to vanquish death and sin. Two, God is a triune God, meaning one God in three persons: Creator (father), Savior (Jesus), Holy Spirit. Yes, it's confusing. Most of us don't get it either. Suffice it to say math does not exist in Christian theology. Three, baptism (the act of sprinkling, pouring, or immersing in water, invoking the name of the Trinity) is a necessary act for initiation and possibly salvation (depending on who you talk to). Four overarching commandments include, "Love the Lord your God with all your heart, soul and strength and love your neighbor as yourself." And finally, for most of us, our major holidays are Christmas (Jesus' birth) and Easter (Jesus' resurrection).
That's a very brief summary, but this includes all branches of Christianity. Now, what about Catholics and Protestants and the rest? They are all branches from the same tree, so to speak. Christianity can actually be divided into three main branches, the Catholics, the Protestants, and the Orthodox. Catholics place an equal weight on the authority of the Bible, the church and oral tradition and we experience God through Sacraments, actions that channel God's grace (baptism, confession, communion, etc), as well as through individual prayer. The Orthodox are actually very similar to Catholics except that we split over beliefs on papal authority (authority of the Pope) and they do have different attitudes about sin and such (sin as crime vs. sin as sickness). Protestants split much later, again over issues of papal authority, issues of corruption, and the desire to make Christianity more accessible to the masses. Note, this is a gross oversimplification of two thousand years of history. I can do more in depth blog posts if peeps are interested.
OK, now, why do some Protestants identify as strictly "Christian" while Catholics are likely to identify as Catholics? I think that has more to do with our history in the U.S. Our nation was and still is mostly Protestant so, whenever people talked about Christians, they were speaking of Protestants. Even today, when you see literature and other products marketed to Christians, they're often using Protestant translations of the Bible and Protestant theology. When Catholics first came here, we weren't considered Christians. Our depictions of the saints and our beliefs about certain rituals (Holy Communion) had other Christians thinking we were gross idolaters. Further, while American Protestants may have had prohibitions actions like consuming alcohol or dancing, Catholics did not (we'd lose all the Irish, Italians, Spaniards, etc;-)). As such, we were seen as immoral. Finally, in my area, people don't really know that much about Protestantism, so it's probably easier for our Protestant friends and family to say that they're Christian, just not Catholic. Why no mention of the Orthodox? There just aren't that many of them, except in areas with huge Greek/Middle Eastern(Christian)/Eastern European Christian populations.
Sorry for the long length! Just thought I should cover this :D