The information here is correct on Jul 31, 2019.
Note that any study involving the number of balls in an individual innings would be inaccurate, as full recording of balls faced was not always done up to the 2000s.
We first take up the King Pair (or golden pair) where the batsman is dismissed off the first ball of each innings). It is theoretically possible to have a Diamond Pair where one is dismissed without facing a ball in each innings. While there are some diamond ducks, there are no diamond pairs recorded in Tests.
First the King Pairs:
The last entry was by Bangladesh’s keeper Nurul Hasan in 2018. Other points of interest can be seen in the above table.
Note the presence of recognized and semi-recognized batsmen such as Richardson, Agarkar (?), Gilchrist, Omar and Sehwag.
Next, there are the “Queen Pairs” or silver pairs:
The number of such instances are 21, just the same as for the King Pair. Here we have two entries by GD McGrath, with the last entry by NZ’s TA Boult in 2018.
The recognized and semi-recognized batsmen include OG Smith, Mudassar, Samuels, de Villiers, Babar and Broad (?). Broad is the only one to get a queen pair or king pair at Lord’s.
Then comes the Jack Pair or bronze pair:
Vettori, Griffith and Powell can be called recognized batsmen. The number of such pairs goes down to 7, with Powell as the last instance in 2017.
And finally, the pairs with 4,5 and 6 balls in each innings:
Together they account for only 6 pairs. They do include recognized batsmen such as Gibbs and DS Smith.
Update: The closest approach to a Diamond Pair appears to be that of Taufeeq Umar (Pak vs Aus, Sharjah, 11-10-2002) who got a silver duck in the first innings and a diamond duck in the second innings.