Yes from BER, it was an IC, but we didn't actually need to go to another city, it just happened to stop near where we were going in Berlin and be the fastest way to get there (first option that navigation apps suggested).
Ended up fine, just got on some slower train that was fine for those tickets.
I'm pretty sure any train from BER airport would be covered under the 9-euro ticket. The trains not covered under the ticket are primarily ICE (inter-city high speed) and anything operated by another company (e.g. FlixTrain).