Can you post a screenshot of your Plan view over here?
Can you right click the second master page, and ensure that it is set to No Master?
Here's what happens when I paste this into a new (no master) master in the main site
It takes on the menu items of the Main Site (except the colors)
Do I have to do the menus a different way in order to keep the different items for each area?
From what I can see you have applied the A-Master to the second master page. Can you right-click the second master page and choose No Master?
But I didn't apply the A-Master to the second master page. And I can right click and it definitely says No Master. It says that under each master as well. I'm perplexed why it changes my menus to the A-Master when pasting something different into a "NO MASTER" B-Master.