Would saving money outside of banking system be able to reduce monetary base? First of all, here is definition of monetary base taken from Investopedia


  A monetary base is the total amount of a currency that is either in general circulation in the hands of the public or in the commercial bank deposits held in the central bank's reserves.


Thanks to this definition we can see why the Fed is able to reduce monetary base by selling bonds. Money that the Fed gets from selling bonds fall to neither category (the Fed isn't the public, money are out circulation and although they can be deposited in Fed's vaults, they belong to Fed and not to any commercial bank), thus from the point of view of this definition these money just disappear. 

It makes me wonder if just taking money from circulation (and consequently - from banking system. Because otherwise money will still "work") will decrease monetary base. Examples:

1.A person named Alex for some reasons keeps all their savings under their mattress

2.Corporation "MacDuck Industries" for some reason keeps its savings in its own building called "money bin". It doesn't loan them, it just sits on its money.

3.The government of Maxtopia has proficit budget and for some reason keeps its excess tax money in its own "money bin". Again, like MacDuck Industries it doesn't do anything interesting with the money, it just sits on piles of cash.

If I understand everything right (including the definition), in all of these cases money are taken from circulation, thus reducing monetary base in all cases. Am I right?
Options 1-2 do not reduce the monetary base. “Money in circulation” are all notes/coins that were issued to the private sector and not returned. 

There is no way of measuring how many transactions notes/coins support (or whether they have been destroyed). So reducing “circulation” has no effect on published statistics.

Option #3 would reduce the monetary base if the money is returned to the central bank.