For me you should only discard one card. Otherwhise it will be written on the base : "After each time a minion is played here".
Tar pits : After each time a minion is destroyed here place it at the bottom of its owner's deck.
Cave of shinies : After each time a minion is destroyed here, its owned gains 1VP
Mountains of Madness : After each time a minion is played here, its owned draws a madness card.
And there is a lot of bases like that.
There are a couple of bases that specify that the ability happens once per turn as well so I wouldn't just assume that the ability only happens once per turn unless stated. I have always played that it happens after each minion but your interpretation may very well be correct too.