One that interferes with a weave from a distance and a separate one that unravels weaves as Cyndane mentions below:

WH
With the CK

Just as it reached the woman, almost close enough to singe her garments, the web of Fire unraveled. The woman did not do anything; the net simply came apart! Cyndane had never heard of a ter'angreal that would break a web, but it must be that.

It would make sense that the ter'angreal Cadsuane used to disrupt Semirhage's MOM was different, or more active, than the passive versions we've seen on Alivia and Mat.

Note the difference in the Semirhage scene:

KOD
A Plain Wooden Box

“No,” Cadsuane replied grimly. “But I can do something about it. Once we’re closer.” Her golden hair ornaments swayed as she tossed her cloak back as though unmasking a sword.

...

Abruptly, no more than ten paces away, she… flickered. For an instant, she was taller than most men. garbed all in black, surprise on her face, and though she still wore the veil, her head was covered with short-cut wavy black hair. Only an instant before the small woman returned, her step faltering as she let her white skirts fall, but another flicker, and the tall dark woman stood there, her face twisted in fury behind the veil. He recognized that face, though he had never seen it before. Lews Therin had, and that was enough.

This seems different than the weave breakers. Even if the flicker was just Semirhage rebuilding the MOM in an instant, the affect was from 10 paces away and affected a weave not directed at the wearer, which we've not seen from either Mat or Nynaeve's ter'angreal. This seems more like a device that slices a weave from a distance at the wearers command.

It's possible that LTT was referring to the defensive ter'angreal, which is supported by Cyndane being unaware of it, rather than the more offensive device Cadsuane seems to possess.

Perhaps the original Paralis-Net included:

Distance Weave Disruptor (that Semirhage encountered)
Well
Angreal
Mood detectors (Nynaeve's set can tell when someone nearby is angry, or has a strong emotion IIRC)

Breaking Era could have included the defensive disruptors like Mat's and the Channeling detectors Demandred mentions.