Do they want us to chase ?

I cannot answer for your particular ex but a lot FAs do it because they are used to the push-pull cycle that was a staple for them growing up.

I did too, before I started therapy. Now I usually stay and don't rush to leave. I talk about things and try to understand. If you want to understand why they do it, there's a number of threads on these boards that explain it. They are recreating the chaos that existed when they were children because that is what they associate with love. That it is not healthy and oftentimes it is not even love is lost to the FA.

I cannot speak for what you should or should not do to win back your ex but I will say what most of the old-timers here say - what do you want? A relationship full of chaos that never settles or a relationship where differences are resolved calmly and one where your partner is not flighty and unpredictable? If chased, an FA settles because one push-pull cycle ended. But it will start again once closeness/intimacy/disagreement/resentment/any problem that requires communication sets it.

They don't deactivate because they want you to chase them. They deactivate because their fear of intimacy and engulfment has gotten triggered and the only way they know how to stop it is distancing. It's more likely they don't consciously recognize or understand this and are just reacting.

Chasing may on some level prove you won't "abandon" them for their bad behavior and allow them to return later, but it also leads to endless testing of you and endless instability because the issue is on their side and will repeat and circle and cycle.

If you are AP or FA but feel stable in other types of relationships, as you said, this is because you're not getting triggered by other types of partners. But the underlying attachment style is still there and will dictate your coping mechanism patterns when stressed, and to some extent also will influence being more attracted to other insecure style partners. I know you're asking about her not yourself, but the most productive response for you in this situation is learning from it and digging more into your own attachment style. You deserve better than a fantasy-turned-breakup. You may not see this, but the drama she was having with the other guy is coming from her (it takes two to continue it) and reflective of her past chaotic romantic relationships and predictive of how she'll act in the future as well. So why would you tolerate that? What were you getting out of a 4 month period filled with projection that isn't otherwise being satisfied in your life?