I just got some "Call fiat First" key ring thingies last week and recall mixed reports on the service on here, so it will be useful to know how you find them.
Putting my lawyer's hat on, I'd also just add that I wouldn't be unduly worried about needing to prove anything to do with the scampering parked vehicle - it is all done on "blanace of probability" and logic dictates that you would not have done an emergency stop for no reason. Added to which, although there are circumstances in which the car in front can be to blame when hit from behind, unless the lady behind you starts lying her head off, that won't be the case here.
I would, however, just jot down and date and sign a note of what she said to you at the time and what she has said subsequently (if anything). I'm sure she is a very nice lady, but you never know, and judges (and therefore insurance companies who know they have to face judges if things don't get sorted) love "contemporaneous written evidence" as we call it. If she has texted you apologetically, save them or print them (if you can).
Glad you are all well. There but for the grace of God go all of us.
