Both the maneuver and count condition must be met.
The steam jacket directly reduces the damage of a move, according to the percentages above. It is currently unknown what damage types it works with. It has been verified to reduce the damage of magical special attacks, as well as spells. Unlike the analyzer though, it didn't work on 1000 needles. It will not "count" a move unless the puppet is harmed.
The steam jacket can only keep track of one damage type at a time. If the puppet is hit by a different type, all memory of the previous will be lost. Thus, this is only effective when the puppet is hit repeatedly by one, and only one type. The count is independent of which mob attacked the puppet, and the puppet does not have to be deployed. Memory persists across multiple mobs, but will reset upon zoning or reactivation.