The bell curve and Gaussian curve (and normal curve) are almost always synonymous. Bell Curve describes its shape, while the Gaussian curve describes the guy who derived that nasty-looking pdf. While some other distributions may look bell-shaped (especially ones that converge to a Normal in some asymptotic sense, like the t-distribution), people will look at you funny if you refer to anything other than the normal (or in rare cases, the t) as "The Bell Curve".
The point about "the data is laid out so the sum of the data adds up to 1" doesn't make much sense since the Normal takes values on the entire real line. Could you clarify?
The procedure you describe is not what most people would call "normalizing", so much as breaking the data up into quantiles/percentiles. Normalizing the data would be for the instructor will subtract the average from everyone's raw score, then divide by the standard deviation. If the raw scores themselves were distributed normally, then these new "normalized" scores are now standard normal, so letter grades are given based on how far away from 0 a student's scaled score is. This procedure is pretty flawed since scores are usually not super close to normality (many exams are not symmetric in scores, the distribution may have several modes, etc). The procedure you describe makes no assumptions about the underlying data, and does not require normality, so I like it more.