Hi, the acidity of methanol, water and glycerol are all very similar – within an order of magnitude of each other which in acidity terms is a very, very small difference. Any of these species can act as acids here. I don’t know your exact conditions but if you were using MeOH as solvent then its concentration will be >> the concentration of any other possible acid in solution.
If you drew MeOH as the acid you would largely be correct; it doesn’t really matter since all protons are identical anyway and there will be reverse acid-base equilibria occurring in solution between all the hydroxyl groups.

I should note that even in neutral water, the protons on H2O undergo very fast exchange with other H2O molecules. Protons are very dynamic. If you ever take an NMR spectrum of an alcohol or a carboxylic acid you will note that these peaks tend to be quite broad, which is due to their relatively fast motion compared to more static protons such as C-H that do not undergo these kinds of exchange processes.

Hope that’s helpful – James