This is a transcript of what I hear from the LiveATC recordings. Times are approximate, UTC.
Thanks so much for taking the time to do this, the liveatc links will be gone after 30 days.
Of note - in the radio calls after takeoff, the crew sounds slightly behind the curve. Tower has to remind them of their departure heading 105. On the Departure tape, they fail to check in and Departure has to call them with "radar contact" and ask for their current altitude.
Sounds to me like they were shaken up just a bit, probably from seeing the runway end rushing at them and taking evasive action, not because they suspected a collision, necessarily.
They were originally filed for the VALLY 2 departure, I'm thinking it may have been changed by ATC to the MIAMI 3 departure to avoid a conflict with departures off 8R, there is no heading 105 on the VALLY 2.
The tower clearing them for takeoff with an assigned heading would be normal for the MIAMI 3 departure, it's 'Climb on assigned heading for RADAR vectors to assigned transition. MAINTAIN 5000', or assigned lower altitude. EXPECT filed altitude 10 minutes after departure.'
I would guess that they were on the PADUS transition since like the VALLY 2, it goes over JANUS intersection.
But you are right, whoever was on the radio for QR 778 was having a little trouble verbalizing, maybe even before the takeoff when they (possibly, it's hard for me to tell) said they were going to hold short of T1 on runway 09, think they meant hold short of runway 9 on T1.
And QR 778 later mistakenly readback the 16,000 climb clearance as 9,000. It was a busy frequency and even the controller had to correct himself on an earlier altitude clearance for QR 778.
As peekay4 points out, the recordings are scanning multiple frequencies so some of the transmissions are inevitably missed or truncated. This might explain why we don't hear QR 778 checking in with the departure controller.
It does appear that QR 778 and the tower both thought they were taxiing on S, not T from the radio transmissions. A landing aircraft was instructed to taxi behind QR on S as well.