The two sets of assumptions are not identical, but do overlap. Here are the assumptions that are relevant to t-tests on means (not all t-tests are on means - e.g. a test of correlation)
1) Observations are random variables that are independent and identically distributed (iid).
This is partially addressed in your class's assumption number 2 (the random part). The random part is not clearly stated in the wikipedia list, though see below regarding identically distributed.
2) The observations are from a normally distributed population.
That the population is normally distributed is listed in both sets. The subtle difference between the two is that the wikipedia version states that there are fixed parameters of mean and variance - implying that the population mean and variance are constant from one observation to the next (identically distributed - as per point 1). This is probably meant to be implied but is not explicitly stated in your class's equivalent assumption.
3) If more than one population is sampled from, then the populations have equal variances (homogeneity of variance).
This is not stated in either list. In the case of the wikipedia example it could be because the formula is written for the single-sample case. You could consider this simply an extension of point 1, but I often see it separately listed.
A) The third assumption you list for your class stuff is partly incorrect. While it is true that larger samples provide some robustness against non-normality, it's not true that they protect against biased sampling. If your observations are not a random sample from the population, I don't think you can fix the problem by taking a larger sample. In any case, the fact that larger samples may be better in some circumstances is not an ASSUMPTION of the test. At best it's a property.
B) Wikipedia's second and third assumptions are ones I haven't seen before. I believe that these are insured in the case of the t-tests on a mean by virtue of the other assumptions I listed (in particular the chi-squared nature of s-squared arises from the assumption of the normality of X in the means tests).