NeverarGreat said:
What if Han’s freighter was stolen from the First Order?
That would make Finn’s assertion something other than completely wrong, and it might just give the scene some suspense.

I think adding the First Order symbol a bit excessive in terms of suspense. I remember when I first watched it, my immediate thought was, “Oh no, the First Order caught the Falcon!” so that suspense already is there.
Having it where Han stole a FO ship just opens a can of worms that I don’t think you can fanedit and justify.