Style of referring to a cardinal As I grew up, I was accustomed to referring to a cardinal by inserting his title between his first and last names. Thus, for example, I was confirmed by the then-archbishop of St. Louis, John Cardinal Carberry. This was distinct from the usage in the case of an archbishop (so, for example, the next archbishop of St. Louis, Archbishop John May).

More recent usage that I've seen indicates that this usage seems to have been discontinued, or at least fallen out of favor.. The official biography of the current archbishop of Newark refers to him as "Joseph Cardinal Tobin". But other carefully written sources such as the American Catholic magazine Crux or the political journal The American Conservative use the phrasing "Cardinal Joseph Tobin".

Is there a usage preferred or designated by the Vatican or a more local group of bishops? Or does it depend on a given style book?
How does one address a Cardinal?

When I grew up, we always referred to a cardinal in a formal way with the salutation as follows (as an Example): His Eminence, Joseph Cardinal Tobin. In the seminary this would have been the way to address a Prince of the Church. In fact this is still quite popular and I still employ it. 

In an article from the Catholic Encyclopedia which deals with ecclesiastical addresses admits that other countries do have varying rules on the subject of how to address ecclesiastic dignitaries and cardinals are no exception. The Article starts with the rules which apply to Italy only and then moves on to other Catholic nations.


  It is from Italy that we derive rules as to what is fitting and customary in the matter of ecclesiastical correspondence. These rules the different Catholic nations have adopted with greater or lesser modifications, according to local conditions, resulting in differences which will be here dealt with. 


Since the article in now somewhat outdated, I will use a more up to date site to show the differences in addressing a cardinal in English speaking counties in our own times.


  A person would greet a Cardinal, for instance Cardinal Keeler of Baltimore, by saying, "Your Eminence" or "Your Lordship" (which is very British). In addressing a letter to Cardinal Keeler, one would write, "His Eminence, William Cardinal Keeler, Archbishop of Baltimore" with the salutation, "Your Eminence," "Most Eminent Cardinal," or "My Lord Cardinal." Lately, some people will reverse the word order, saying, "Cardinal William Keeler" instead of "William Cardinal Keeler." The formal word order originated in the time when last names were not common, but individuals were known by occupations or even places. For example, "John, the Smith" (or Blacksmith) eventually became "John Smith." The same evolution occurred with Cardinals: What would have been "William, the Cardinal" would now be, with the use of family names, "William Cardinal Keeler." - How Do We Address Church Officials?


It is interesting to note that during the the Church's Habemus Papam! ("We Have a Pope!") ceremony, the Cardial Protodeacon  uses the former method in annoucing the newly elected pope:


  Habemus Papam! ("We Have a Pope!") is the announcement given in Latin by the Cardinal Protodeacon, the senior Cardinal Deacon, upon the election of a new Roman Catholic pope. 
  
  The announcement is given from the central balcony of St. Peter's Basilica in the Vatican. After the announcement, the new pope is presented to the people and he gives his first Urbi et Orbi blessing. 
  
  The format for the announcement when a cardinal is elected Pope is:
  
  Annuntio vobis gaudium magnum:
   HABEMUS PAPAM!
   Eminentissimum ac reverendissimum Dominum,
   Dominum (first name) Sanctæ Romanæ Ecclesiæ Cardinalem (last name),
   Qui sibi nomen imposuit (papal name). 
  
  In English, it can be translated as: 
  
  I announce to you a great joy.
   WE HAVE A POPE! 
   The Most Eminent and Reverend Lord,
   Lord (first name), Cardinal of the Holy Roman Church (last name),
   Who takes to himself the name (papal name).
  
  Habemus Papam (Wikipedia)


Here is the 2005 Habemus Papam of Pope Benedict XVI found on YouTube: Habemus Papam: When Cardinal Medina introduced the new Pope.

I personally like traditions and on that note I would like to add the following: As with a Patriarch, you should stand when he enters a room in which you are present (until he invites you to sit) and again when he leaves it. Remove your hat in his presence, and you may kiss his ring during a greeting and/or as he leaves. You are free to kiss his ring out of respect for his office. You may kneel when kissing the ring (though bowing at the waist is also acceptable); however, do not do either if the Pope is present.