I did not say Aragorn did not know it was a Balrog until Lothlorien. Aragorn is reporting the events that took place in Moria and is clearly saying that when he saw the creature he did not know what it was at that time. I of course assume Aragorn heard Legolas and Gandalf identify the Balrog.
I find it hard to believe that, when he heard the story of what happened in Moria in TA 1980, that Gandalf didn't say to himself "hmmm.....walks like a duck, talks like a duck, kills like a duck....". I can only imagine that after the Balrog goes nuts in Moria that there was incredible destruction and, since there were no further incidences with the Balrog, that the Wise thought it was "imprisoned" again....or left.
You may find it hard to believe, but that is the way Tolkien wrote it. As for the Durin's Bane/Balrog having left Moria, is it likely that something so terrible that it could drive off the dwarves from their beloved city would be able to slip off quietly in the night without anyone in ME knowing about it. Yes, Gandalf and the wise could have thought Durin's Bane (whatever it was) could have gone back to sleep under the mountains, but it is unlikely that it had left Moria. The Watcher in the Water seems to have been an unknown terror, unknown even to Gandalf. This shows that not all terrors were known even to the wise. Durin's Bane could have been thought to be another of these unknown monsters.



