You did find a very nice example online that fails to understand the definition of the term pKa along with a certain equilibrium that will be at play when you mix a base with an acid other than its conjugate acid.
Methoxide is derived from methanol, pKa 15.5. Water has a pKa of 15.7, so we can expect exchange of protons under these conditions.
MeO- + H2O ------> MeOH + HO-
I'd be careful with this problem. Just because it's on a UC-Davis website doesn't mean your professor believes it's correct. The area of SN1, SN2, E1, E2 is where authors and professors <<<disagree most>>>.
What I'd also like you to realize is that this flowchart also came from University of California, albeit in Los Angeles.
http://voh.chem.ucla.edu/vohtar/spring06/classes/30A/pdf/Flowchartfor.pdfAccording to the chart, the product for this reaction is one of E2 elimination. Also, Vollhardt & Schore (Berkeley) indicate this as E2.
For this particular topic, what I or anyone else other than your professor tells you is irrelevant. Your professor grades your exam. It's in your best interest to talk with him/her about it - NOT your TA.