PSA was developed and published later than PISA, and had as one of its goals to be precise, and include most or all of the details visible to a P4 developer that you would want in order to answer questions like “what does multicast replication do, and what packets show up in egress as a result?” and “what does cloning/mirroring do, and what packets show up in egress as a result, with all of its headers and contents, down to the last bit?”
If I recall correctly (I haven’t re-read the source you mention by Hauser et al recently), PISA, by contrast, is described at a higher level, with fewer details, and is not sufficiently detailed to answer those kinds of questions. This is not a criticism of PISA – it was written with a different purpose in mind. Those details would have been a distraction from the purpose of the paper.
TNA was initially developed as a proprietary architecture by Barefoot and later Intel (after acquiring Barefoot), with its details very well fleshed out, but only given to people who had signed an NDA (Non-Disclosure Agreement), and not published in more detail until 2021, here: Open-Tofino/PUBLIC_Tofino-Native-Arch.pdf at master · barefootnetworks/Open-Tofino · GitHub
The Tofino ASICs always had a hardware deparser before the traffic manager, but the P4_14 language plus architecture did its best to hide its existence from the P4 developer, with the P4_14 compiler for Tofino configuring the ingress deparser hardware so that the P4 developer often would not need to know of its existence.