Everything listed there you would have found during the frontier days of America (and still do in some parts), so does that make America not western? I don't know it's very confusing. My parents are from Greece from large families and had this exact same experience growing up -- you had to do your part -- and Greece is known as the the birthplace of western civilization.
Again, confusing to me. Western is just some huge bucket now that doesn't really have meaning.
On another level, I can kind of relate to you not knowing what type of "western" you are. On a whole, (this might sound confusing) it's the Roman West that is overrepresented in "Western Civilization", but you also had the "East" western civilization as well, and very much has some things in common with what some people know was "East" (as in Asian), but still distinctly "Western"
But I am with you, I don't know either sometimes.