An attack "action" is a Creature using a full or quick "action" to make an attack... see page 3 of codex. An attack sequence is the numbered steps through which an attack progresses. They are not exactly the same thing, though you can only have an attack sequence if there is an attack action.
Page 29 under Battle Fury, the Codex says "If the extra attack is made against a different Defender it is treated like a Sweeping Attack, and starts a new attack sequence...." Therefore, each attack made with either Battle Fury or any form of Sweeping (such as Whirling Attack), is a separate attack sequence.
The second paragraph under Sweeping shows that there is one attack action (though two attacks) made by a creature with the Sweeping trait.
Also from Attack Actions on page 3: "... and Sweeping are one attack action, but consist of multiple individual attacks." I understand this as to say that each attack using the Sweeping trait progresses through a separate and complete attack sequence, while only taking a single Full Action Attack. Note that the chart on page 3 shows that there are three separate attack sequences (as step 8 says to "repeat above" which starts with declare attack... you do step out of Step 8 when you go back to Step 1 again by the various notes above.
Since you start a new attack sequence when performing the second attack using Whirling Strike, but are not beginning a new attack action... Whirling Strike should be resolved in full before the Battle Fury attack takes place; the (Full) action is not completed until the final Sweeping-like attack is made.
Battle Fury: "The next time this round the target creature makes a melee attack... at the end of that attack action it may immediately make another quick action melee attack. This extra attack counts as part of the same attack action...."
This means that the Battle Fury starts a new attack sequence within the same Full Attack Action, but this attack is not restricted to a target that has not yet been attacked, like the others in Whirling Strike.
Whirling Strike: "Once this round, if this creature uses a full action to make a quick action melee attack, it may use that attack against up to 3 different target objects in its zone. It attacks each target, one after the other, in the same manner as the Sweeping trait."
So... attack sequence is not the same as attack action, both cards make perfect sense, you must go through the Whirling Strike in its entirety first, and then Battle Fury kicks in at the end in the same attack action.