Is the Forager's coin bonus counted during Action Phase or Buy Phase?
It gives you the coin when you play it. You just don't use it until the buy phase.

This is really a case of "do what it says on the card, in the order it says it", what you called "top to bottom" in your question. It's true that a lot of people tally up their coin at the start of the buy phase, but that's just because it's when they care about it, not because the coin didn't exist until then. As the original rules say:


  +$X -- the player has X number of additional coins to spend in the Buy phase


They're granted immediately, and available to spend later in the Buy phase.

Similarly, if you play a Salvager and trash something costing $4 to get +$4, then play a Bridge, it doesn't retroactively make the Salvager only worth +$3. And if you play a Bank, it only counts the treasures that came before it - you don't get to wait until the end and count everything for them.

You can see hints of this elsewhere, like in the wording of duration cards. For example, Merchant Ship says "Now and at the start of your next turn: +$2." It grants the coin at the start of your next turn, not on your next turn's Buy phase.