[Lorca]had no way of knowing Starfleet would give him another ship after his last one was destroyed. Better to try and fool everyone into thinking he's Prime Lorca - at least initially.
Maybe his cover was blown, and he was forced to destroy the ship. But I think he'd only do that as a last resort. Honestly, the timing would be difficult, because he'd have to do it without letting the crew know and send out a broadcast to Starfleet. It's much easier to just assume the Klingons destroyed it.
I'm inclined to agree with your interpretation. But we really shouldn't have to be speculating about this... for a storyline as major as this was, the necessary backstory should have been provided in full, rather than just shoehorned in via a ten-second flashback with an ambiguous voiceover and bad SFX.
(Hell, we got Burnham's mutiny backstory at
painful length, spread across two (boring, murky) "prologue" episodes, when it could and should have been parceled out via flashbacks over the course of the season. I'm coming to the conclusion that these writers really have no sense of pacing.)
Burnham worked out the whole thing speaking out loud right in front of the Emperor. She announced the conclusion of her thought process regarding Lorca's true identity to the Emperor when she stated, "he's not from my (our) world, he's from yours". The Emperor's ensuing silence was confirmation that Burnham had reached the correct conclusion. Had Phillipa not agreed and thought Lorca was actually not from her world she would have said so.
And, so what? The Emperor had no way at all of knowing if it was true either. She hadn't exchanged so much as two sentences with Lorca before having him thrown in an agonizer booth.
By continously lying to Burnham, Lorca had destroyed any credibility, so why would Burnham trust him to tel the truth? Phillipa had shown herself to be brutal, dictatorial, etc, but she had not yet lied to Burnham yet, so... Burnham throws in with Phillipa.
Lorca hadn't "continuously" lied to Burnham, he'd just told her one Big Lie, in order to protect his universe of origin. (Just as she herself was doing on he MU
Shenzou, as others have pointed out.) Beyond that, she knew nothing of his motivations.
As for Georgiou, how on earth could we or Burnham or anyone say "she had not yet lied"? What basis did Burnham have to test the truth of
anything about her situation? Certainly there was nothing about the Emperor to suggest she'd have any scruples about lying, and plenty of what she said to Burnham in ep 12 seemed pretty suspiciously like manipulative gaslighting to turn her against Lorca.
Honestly, what I was hoping and expecting to see as of the end of ep 12 was a follow-up in which the Emperor would betray Burnham's (provisional) trust, as she had ample means, motive, and opportunity to do... whilst Lorca turned out to have credible motivations for wanting to overthrow the Emperor (not necessarily noble ones, but at least more complex than self-aggrandizement and xenophobia), thereby explaining his devotion to the long con and the loyalty of his band of followers... and Burnham was torn between the two of them, tempted to judge a book by its cover (Georgiou) and give in to feelings of personal betrayal (from Lorca), but in the end realizing that the man she's served under was at least a better bet than a mass-murdering cannibalistic psychopath.
Instead, what we got was a story where she not only gave in to that temptation, but it turned out (fortuitously and implausibly) to be the right thing to do(!). I was more than a little disappointed. Frankly, I can't help but wonder if Ted Sullivan (who wrote ep 13) even
read ep 12, because thematically speaking the two seemed to clash pretty glaringly.
I just re-watched and she pretended to go in with Lorca in an attempt to save Discovery after her explicit summation that Lorca would let D's crew die and/or use the ship to invade the Prime Universe. She threw in with Georgiou to get to the throne room so that she could drop the containment field around the Charon's spore engine thing. All the information is there.
Burnham's summation didn't qualify as "information"; it was sheer speculation on her part. As for her "plan," such as it was, it was a hail-mary pass of the most extreme kind, and her decision to trust Georgiou, even a little bit, to help with it, was inexplicably foolhardy (as was Georgiou's decision to cooperate and provide that help).
Bottom line, episode 13 has really, really soured me on this show. After the kind of buildup and anticipation we've had, you've got to nail the landing, and the writers completely failed to do that. Okay, it wasn't as big a disaster as (say) the wrap-ups of
NuBSG or
Lost, but then it wasn't a series finale, either. Still, it pretty much exhausted my patience and goodwill where the show's writing is concerned.