How to close a tab in vim?

You can use :tabclose.

Read more with :help tabpage.

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As for the buffer coming back with :bn, I believe this is normal for "hidden buffers", and :bn wraps around to the first buffer when you go past the last one.

See :help buffer-hidden

You're mixing up tabs and buffers. A tab is simply a way of displaying buffers, closing it doesn't do anything to those.

If you want to delete a buffer from the buffer list, use :bd. From the help:

:[N]bd[elete][!]            *:bd* *:bdel* *:bdelete* *E516*
:bd[elete][!] [N]
        Unload buffer [N] (default: current buffer) and delete it from
        the buffer list.  If the buffer was changed, this fails,
        unless when [!] is specified, in which case changes are lost.
        The file remains unaffected.  Any windows for this buffer are
        closed.  If buffer [N] is the current buffer, another buffer
        will be displayed instead.  This is the most recent entry in
        the jump list that points into a loaded buffer.
        Actually, the buffer isn't completely deleted, it is removed
        from the buffer list |unlisted-buffer| and option values,
        variables and mappings/abbreviations for the buffer are
        cleared.