Who does the Runner pay if a piece if ICE states "runner... pays X Credits"?
All transactions are conducted with the bank, credits never change hands directly from one player to another.

Thematically, this makes sense as you can think of runner and corp using different scales of money. One credit to the runner may represent say $1,000 or $10,000, but for the corp it may represent 10 or 100 times that amount.

From a game design perspective, it makes sense as the game was intended to be played with the cardboard credit tokens, but also designed for play between people who own their own sets. With neither player directly giving money to the other, there's no confusion as to whose game pieces are whose.

I believe the most specific place in the rulebook is the Credit Pool definition on page 6:


  Each player has a credit pool where he keeps the credit tokens
  he has available to spend. Spent credits are returned to the
  token bank.


"Pay", "lose", and "spend" have the same effect of credits returning to the bank. The difference is that payments and spending are often optional (unless preceded by "must"), and the payer gains something. "Loses" is common as a penalty from the other player, e.g., from an Account Siphon.