Elayne intended to use the Band and the rest of her armies against Perrin only as a last resort to suppress an attempt to revive Manetheren and take over half of Andor, or even try to take over te whole of Andor. As far as she knew when she first met Mat in ToM, this was a distinct possibility since Perrin had raiased a big army, raised the Manetheren banner and also had this fight with Rand before he left for Ghealdan (she doesn't know this was staged). As far as she knew before meeting him, Perrin's forces may have had no intention of participating in the Last Battle and raising them to take over part of Andor is just as likely.

As for Cairhien, she tried to take over it peacefully and it cost us quite a bit of of her personal holdings, which she would have to use to compensate those Andoran nobles which lands were used to convince the Cairhienian nobles to support her. This doesn't strike me as an action of someone who cares only for her personal power and Andor's expansion.

My reading about the mentioning of the Band "as part of my army" in Cairhien is that she intended to use the fact she has the support of so many Cairhienian soldiers (including quite a few nobles) as a tool to convince the other nobles there to accept here easier and with minimum fighting, preferably none at all.