The triggered ability can trigger only once each turn. It doesn't matter whether Foggy Swamp Spirit Keeper was on the battlefield when the first card was drawn. If it's not on the battlefield when the second card is drawn, the ability can't trigger at all that turn. It won't trigger when the third or fourth card is drawn.

If a card or ability instructs you to put cards into your hand without specifically using the word "draw," it's not a card drawn.

If an effect instructs you to draw multiple cards, the ability triggers after you draw whichever is the second one for the turn. It goes on the stack after you've drawn and looked at all the cards and finished resolving the spell or ability that caused you to draw them.