Assault keyword appears on schemes. When a character a basic thwart on a scheme with the assault keyword, that character has to use their ATK instead of their THW stat for that thwart. If that character is an ally, it takes the consequential damage listed under its ATK after the thwart. This action is still considered a thwarting action, not an attack.
The assault keyword was first introduced NeXt Evolution expansion.
Do attachments that increase (or decrease) my ATK stat affect the thwart value?
Yes. Any attachment, or event that increase (or decreases) your basic ATK stat, is considered when deducting threat from a scheme with assault.