You got the right of it with your edits about “both”. Foster is a variant of “Announce future badness”, and the ambiguity is an intended feature. As a GM, you have those two moves that drive you towards the idealized dramatic situation. Threaten asks you to think of the procedural situation, which might trigger emotional reactions. Foster asks you to think of an emotional situation which might have a procedural context. The players will re-contextualize, determine how they respond; either emotionally or procedural. Since harm is mechanically represented by increased stress on one of the tracks, both are mechanically synonymous.

But yeah, I was intentionally trying to pack the GM moves so that they follow a specific formula as much as possible.
1) Prompt action or decision in the fiction.
2) If they fail to take action or make a decision, they suffer the relevant consequences
3) Pass the narrative to the players..