Except, the whole encounter was FUBAR. Starfleet committed war crimes, and fired on a Klingon national. Regardless of what we saw what was reported was a train wreck with massive losses and a mutiny and a war crime. Yeah, it was Klingons wanting war but what information was available to Starfleet didn't paint a pretty picture.
Wait... you've lost me.
How did SF commit war crimes and fired on a klingon national in the first 2 episodes?
Prior to the BOTBS ship to ship fighting actually starting, the only thing that happened was that M.B. killed a Klingon warrior in self-defense on the hull of the artifact (whilst keeping in mind that the Shenzou was originally dispatched to that position to investigate one of SF's relays being destroyed - aka, first Klingon attack).
As for what info was reported back...
Let's see if I get the timeline from SF's perspective here:
1. SF gets notice that a relay stopped transmitting in the binary star system,
2. SF dispathes Shenzou to investigate.
3. Shortly after, the Klingon artifact puts out a massive amount of luminosity and frequency which turned out to be a call to arms (for reinforcements).
4. SF is alerted about the Klingons and dispatches more ships to the Binary system.
5. A battle ensues which results in SF's defeat.
6. The some crew from the Shenzou (incl. M.B.) and other ships are retrieved by SF and debriefed.
7. M.B. is implicated to have staged a mutiny onboard the Shenzou in an attempt to deal with the Klingons how she deems is necessary (but is was at the time unable to execute it because the bridge crew [some of which survived and found themselves on Disco] SAW Burnham being intercepted by Georgiou and throwing her into the brig).
Now, the only thing the crew wouldn't know for sure is whether or not M.B. was being honest when she decribed what happened during heir space walk on the Klingon artifact vessel (which again is moot because they could have been scanning her to ensure that she was being honest - like Admiral Cornwall did when she was interviewing Spock in Season 2).