Hi and welcome, Mary
I believe you can do all of what you want with vbulletin. It's secure and I believe there is an option to manually approve new registrations to control who becomes a member. And if you don't want it visible to the public then there is a feature to only allow members to view the threads/discussions.
As far as design, you have complete control. Most people either don't want to go through the steps to incorporate it into their design or don't know how. It does take some tweaking and adding HTML/CSS to the footer/header templates of the forum but it is completely do-able and the "Admin panel" tells you where to edit the code to edit the header, body and footer of the forum. You can also change the colors of the forums if you'd like.
Vbulletin also has a nice members forum you can use if you want help with customization and other options. It's a very powerful piece of software and I bet I haven't used half of its features.
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