She used to be a passive weapon. The real nasty fighters among the Seanchan channellers aren't the damane, but the sul'dam, who use them in battle and direct what the damane do. The damane just do the weaving, none of the decision making. Their skills cover a fairly limited area of fighting too, as the Seanchan have rather specialized, traditional and limited uses for their damane. They're used to fight as part of non-channelling armies, and be used for specific tasks. They're considered too valuable to put at the frontline, they're usually kept behind the lines, away from the enemy.

The Seanchan in general have fairly little experience fighting people who use the OP against them, barring the occasional rebelling High Lord or Lady who had damane and sul'dam. Things must have been different centuries ago - while the Empire was expanding and conquering "AS" kingdoms, but by the time the Alivia was born, the Empire was already pretty much consolidated. Damane vs. Damane battles still happen, but far more rarely as most damane are owned by the Crystal Throne. Lords who are allowed to own damane don't have a lot of them.

Cyndane had massive advantages over Alivia. She lived and survived the War of Shadow, for one thing, and among the top ranks of the Chosen, who backstabbed one another on a regular basis. It's conceivable that people like Mesaana, Aginor and even Moghedien managed to avoid many confrontations because of their personality, but Lanfear isn't exactly a diplomat. She's confrontational, big mouthed, temperamental and no coward. Just looking at the way she dealt with Asmodean, or confronted Ishamael to his face, that one must have fought quite a few nasty duels through the WOS, with people jealous of her or that she massively pissed off, far more often than she had to fight Lightsiders in person, most likely...

Of course, Lanfear's skills with saidar put Alivia's to shame. How many tricks (inverting and so on) and weaves that can be turned into weapons in a fight Lanfear must know, that Alivia has strictly no clue about? Alivia knows basic battle weaves, and is used to wait for orders to weave them. There's Lanfear's speed to account for as well, and also, the Chosen must be used to face opponents who have angreal and even san'angreal. They were common place in the WOS,at least angreal. You approach a fight differently if you know you're overpowered or matched in raw strength, which Cyndane figured out right away about Alivia (she figured out she was really powerful, anyway). There's also the fact Alivia had zero experience with an angreal: she was just the same Alivia, using stronger versions of her usual basic fighting weaves, and for the first time of her life, without a sul'dam to guide her. Strength is just one factor (case in point: the Chosen remain to fight against mixed gender circles... they were vastly overpowered in strength but still considered fighting...). A lot after that depends on the circumstances, and what you can do with all that strength. Lanfear might not have survived had it been LTT, Demandred, Graendal or Semirhage using that angreal against her, but an angreal doesn't in itself makes you a more skilled weaver, or faster. I would think a Forsaken is more than a match for a damane, even a damane with an angreal - especially not when the only channellers this damane ever faced (beside Asha'man) must have been other damane with similar training to hers. It appears Cyndane after her initial surprise at Alivia's strength managed to hold her own, deflect or split Alivia's weaves despite their strength, perhaps managing to hide or tactically retreat for a while, eventually overcoming her with her speed and far more varied arsenal, and in the end she managed to drive one of her attacks home and put Alivia out of the fight. We've only seen a few of the battles that must have happened that day, and only the beginning of the day for the most part. By the outcome of the Cleansing - with few losses on both sides, it appears the Chosen like Graendal, Cyndane and Demandred managed to recuperate from their initial surprise and after a while, thanks to their greater skills and experience of fighting, managed to put the circles on the defensive. The greater strength of the Lightsiders and their toys prevented the Chosen from achieving decisive victories (but for Cyndane and Aran'gar managing to put Alivia and the circle with Eben out of commission... minor victories), but clearly the skills, experience and cunning of the Forsaken were great enough that the Lightsiders were not able to do much more with their much greater raw strength... merely killing one Chosen not very adept at fighting, who didn't watch his back (and rather obviously, had remained away from the fights for most of the day, not to have figured out by then that the enemy had some paralis nets... what happened to Osan'gar would not have happened to most of the others who got more involved with the circles and knew by then they only had seconds to strike and move out before the counter attack came). They must all have become more clever at facing the situation as the day went on. It appears Cyndane didn't injure Alivia right away but rather much later in the day (if the circle with Eben, and Alivia, had been put out of commission very early, the Forsaken would probably have won in the end), which suggests she managed to escape the initial duel, figured out her channelling was being tracked, and got more clever with surprise attacks (Cyndane has the personality not to give up easily once Alivia pissed her off). Surely, Lanfear didn't duel Alivia constantly all day long.... Some of the Chosen probably managed to Travel outside the range of the "paralis nets" (Those "paralis nets" seem to be a mistake introduced by Brandon (or an unnoticed disparity between the background notes, and details Jordan decided to change as he wrote Demandred's POV in WH. Brandon might have overlooked that their early backstory as a WOS device named paralis net in the notes was obsolete). In WH, Demandred jumped to the conclusion they were made during the Breaking and reasonned out why such a device would have been invented then, it seems to contradict the notion that LTT had one in the WOS, or that Semirhage knew a name for the device. Demandred would rather have said then "it appeared some paralis nets had survived the WOS, and of course every woman with the ability to make ter'angreal would have been asked to create more of them during the breaking" after a while, and returned on foot, not channelling again until they were in range for a surprise attack on a circle.