According to the FIE rules, an attack is defined as "the initial offensive action made by extending the arm and continuously threatening the opponent's target...".
This is explained in the USFA Fencing Officials Commission FAQ: Initial refers to which fencer starts the action before their opponent does.
Threatening indicates that the attacker is in advance-lunge distance and close enough to hit, and their weapon's point (for foil) or blade (for sabre) are approaching the opponent's valid target.
In practice, referees, especially inexperienced ones, may go for the easy option and give priority to whichever fencer happened to be moving forwards.
The adherents argue that this is the more classical way of doing things, but this claim is dubious, as actual practice decades ago based right of way on which fencer started straightening the arm (not which fencer completed the extension); and the reworded rules conform better to actual, traditional practice which was documented in some older editions of the rules.
For example, the 1957 Amateur Fencers League of America (AFLA) rules says The rules do not require that the attack be made with a fully extended arm; in fact, there are outstanding international champions in foil and sabre who attack without extending the arm fully.
Consequently, foilists often parry with a sharp beating motion which does not necessarily end in a guard position that closes a line.