When editing or adding Individual and Organization contact types the main menu doesn't update the contact type text. For example the options under Contacts > Individuals > are still New Student, New Parent, and New Staff, when they should be New Member, New Intern, and New Staff. When I click on "New Student" it brings me to add a new member page, so it's really just the menu list that is not updating and the pages themselves are fine.
Thanks for reporting this and describing it so clearly. I have reproduced this problem on the latest version of CiviCRM, though in my testing I found the problem to be limited to editing the label of existing contact types (adding new contact types work okay in my testing).
This looks like a bug to me. I would suggest you take this a step further by filing a bug report. You can use the text you wrote already and even link to this page too.
In the mean time, if you're still in the process of configuring CiviCRM (and haven't yet created contacts with your desired sub-types), you can delete those problematic sub-types and create new sub-types for "Member", "Intern", and "Staff".