No, Lightroom would have been ok too. I just needed to make sure that you weren't viewing in a non-colour-managed program like Windows Viewer or something like that.
Great. Then that is the one you edit on. Use the other one for emails and web browsing and such.
This is very normal. It's almost completely impossible to make multiple screens match each other exactly, even if they're exactly the same make and model and age.
So talk to me more about the smaller screen. Is it a good match, or is it still too bright?