Heisenberg Interaction Picture reference I recently found out that "the interaction picture" comes in two variants--"the Schrodinger interaction picture" and "the Heisenberg interaction picture". 
I am looking for a good reference on the complete details on how operators and states behave in the "Heisenberg interaction picture".
One famous example of such operator evolution can be found in eq. (2.22) of https://arxiv.org/abs/1208.5174
a) I am trying to find the details on eq. (2.2.3) which is given the name "the Heisenberg interaction picture"! (page 18 of the book preview) https://www.google.com/books/edition/Methods_in_Theoretical_Quantum_Optics/Gw4sxyr6UhMC?hl=en&gbpv=1&printsec=frontcover
b) The relationship between operators in the Heisenberg picture and the Dirac(interaction) picture can be found in eqs. (5.2) or (5.3) here.https://projecteuclid.org/download/pdf_1/euclid.cmp/1103839389 I am trying to work out how a matrix element would be the same in the two pictures.
 A: You are mixing up the terms. The Schrödinger picture (or Schrödinger representation) is a formulation of quantum mechanics in which quantum states depend on time, whereas operators are constant with respect to time. In the Heisenberg picture (or Heisenberg representation) it is vice versa: the operators explicitly depend on time and states do not. Thirdly, in the interaction picture both operators and states depend on time and the most distinctive feature of this representation is that you define the time-evolution operator, which in your case is (2.21).
The interaction representation as follows from its name is useful while solving the problems where Hamiltonian can be separated into non- and interacting parts, which is explicitly done in (2.19). You can see that the first term just gives the energy of modes and the second terms describes two-mode excitations as long as I understand. The (2.22) is exactly the formula from the Wikipedia page of the interaction picture (the definition of operator).
The easy readable textbook reference could be Sakurai's "Modern Quantum Mechanics" for the topic of representations. It is also a great book to read in full, as it covers most of the topics you'd need to study/work on the basic level.
A: With the help of the post Are there more than 3 dynamical pictures of quantum mechanics?
I shall try to prepare a complete elucidation of the two interaction pictures here.
( @jgerber, please help me work out the stuff you left as exercises)
\section{Matrix Elements}
For the observable, O(t), we have
\begin{eqnarray}
\langle O(t) \rangle
\end{eqnarray}
