How do I use and show a vocabulary with multiple fields? When I create a new vocabulary, I can create it in such a way that there are multiple fields. However, when I then create a content type with a field a term reference to that vocabulary, I only can see the name field from the vocabulary and not the other fields. 

This regardless of the method I use, whether be radio buttons, check list, select list, autocomplete, autocomplete deluxe. I want to show those other fields so I can discriminate between two very similar, but with distinct values in the vocabulary. 

For example, if it is a vocabulary of names, to add a date of birth which gets at least shown (and preferably is editable when adding a new term using autocomplete deluxe) when I fill in the form when adding an item of that content type.
What you're looking for is Entity Reference.  This will let you create a taxonomy term field on your content type and give you the option to render the entire term using the display of your choice (teaser, full etc).

Setting it up would be very similar to what you've already tried with term reference:


Create an entity reference field in your content type
Target type = taxonomy term
Set target bundle if preferred, then save
Under manage display, set the field format to rendered entity
Set the view mode to your preferred option (you would configure these options under your vocabulary manage display screen)