What is the chemistry behind cleaning silverware with aluminium foil

This question has a accepted answer on this site already:

What happens during cleaning silverware?

but the reason I ask the question again is because I found different "recipes" for the technique and also different reaction formulas in other sources.

The "recipe" given in the linked question is

"Put [the silverware] into a pot with some club soda and a piece of aluminium foil and pour over a hot water."

And the reaction formulas given for "blackening" is the following

$$\ce{3Ag2O(s) + 2Al(s) -> 3Ag(s) + Al2O3(s)}$$

and for the reaction with the aluminium foil

$$\ce{3Ag2O(s) + 2Al(s) +3H2CO3(aq) -> 3Ag(s) + 2Al(OH)3(aq) + 3CO2(g)}$$

But I also found the following "recipe" in a Swedish newspaper

"Put aluminium foil in a bowl. Put some baking powder on the foil. Put in the silverware. Add more baking powder. Pour over hot water and wait."

I assume this recipe is more or less equivalent to the one given earlier since the baking powder essentially turns the water into carbonated water?

The reaction formula for the "blackening" of the silver given in the newspaper is

$$\ce{Ag + sulfurcompunds + O2 -> Ag2S + H2O}$$

and the reaction with the aluminium is described as following:

$$\ce{2 Al(s) + 3 Ag2S(s) + 6 H2O(l) - 6 Ag(s) + Al2O3(s) + 3 H2S(g)}$$

The newspaper also links to a small piece written at a University in Sweden with regards to this question.

where they explain the blackening of the silver in the same way as the newspaper but they give a different reaction formula for the aluminium reaction

$$\ce{2 Al + 3 Ag2S -> 6 Ag(s) + Al2S3}$$

So my question is: Is any one of these reaction more "true" (occurring more often naturally) than the others or is it the case that a little bit of everything is happening?