The First Jewish History Class

Today we had our first day of Jewish History class, and despite some lingering jet lag I thought it went great. The main question we asked today is a seemingly obvious one; "What is Jewish History?" The "history" part, which we'll examine a bit more in depth tomorrow can be thought of as "the events that  happened in the past". While hardly an in depth explanation, it's at least a meaningful approximation. The other part of the phrase, "Jewish," is significantly more complicated. To try and get a better understanding, or at least a usable definition we did some word association, in which the students shouted out anything that came to mind when they hear the word "Jewish". Here's what they came up with (click on the image to enlarge):

What comes to students' minds when they hear the word "Jewish"

As you can see their responses really reflect the broadness of Judaism (though I'm still not 100% sure what Muhammad Ali has to do with anything). After this brainstorming the students, in chavruta, tried to organize this mess by putting these thoughts into categories. As you can imagine, they came up with many interesting ways to categorize the terms. One of the ways that I found particularly interesting that I had never considered was to categorize them as "secular" or "religious." At the end of class I explained that one common way to think of traditional Jewish identity is through a triangle of three ideas: Am Yisrael (literally the people of Israel, meaning the culture and customs such as Jewish foods, languages, humor, etc.), Toraht Yisrael (literally the five books of Moses, but here more generally including all of the religious aspects such as prayer and study) and Eretz Yisrael (the land of Israel).
Clearly virtually no aspect of Judaism fits neatly into one category. Pesach (Passover), for example, definitely has religious meaning, and we're celebrating an event (the exodus from Egypt) that's mentioned in the Tanakh (Jewish Bible), but many non-religious Jews still sit down for Seder. And while my mother's matzoh ball soup is certainly delicious enough that it may well qualify as a religious experience, I think it probably makes more sense to think of it as cultural (or Am Yisrael) rather than religious (Toraht Yisrael). Similarly, the blessings we say over the candles, wine and bread are religious, but the reason we bless those specific items is because olives (from which olive oil lamps were made), grapes (wine) and wheat (bread) were the main agricultural products of ancient Israel, so the blessings are related to Eretz Yisrael as well. And how about the family that says the blessings for traditional or familial reasons? Should this act then be viewed culturally, rather than religiously? This view of Jewish identity raises many questions. Do you think in modern day America Judaism has cultural aspects? Are they more or less important than the religious aspects? Do you think Jews living in Israel would feel differently?