Do you know the pinout of the Ethernet/RJ45 cable? Do you know what kind of protocol / interface they are using (probably not as that is proprietary information)? That cable has 8 contacts (8P8C). How do you know that the video signal can be completely transmitted using only 3 of these 8 contacts? You are probably receiving digital input, but you want to somehow convert that to analog RCA signal which can not be done with a simple connector.
I think what you will need to do is completely understand the interface pins on the ethernet connector and each pins function. You will then need to design a circuit / board with an application processor code/software that can decrypt the incoming signal, perform the digital to analog conversion and then output it on an RCA interface.
Apart from the legal concerns as stated by Alabalcho, I do not think it would be easy, even if you did find and understand all the protocols / encryption techniques etc. You would require knowledge of all their protocols, a degree in electrical engineering and probably someone who has designed low power high speed circuits before.
I do not think this is worth attempting
Did you ever read the whole discussion about that? It's perfectly clear that that Ethernet cable is not carrying video signals, but Ethernet frames.
I missed that, sorry, that should make things slightly less tricky
Even if OP reads Ethernet 100Base electrical standard, and TCP/IP protocol, and frames, what device the OP would need to decode / decrypt the Amazon Video / Apple TV stream, re-encode it back, and output it on analogue outputs suitable for the recorder?
Like I said he will need to design the hardware and software himself. I imagine this could be done using an ARM based App processor like an iMX with an ethernet physical interface, or perhaps a cheaper microcontroller like an STM32 along with the relevant DAC's and any cryptography HW. If he understands the encryption and protocol he will have to write the code himself to decrypt it and extract the sound / video data to the analog outputs on the target chip, composite out can then be reconstructed in software and brought out to a physical connector.
Of course you may hit many brick walls during development and realize things like "oh I can't do it on this processor, oh I need something faster, oh i'm getting EMI interference I need to make some signal conditioning chains or put ferrite shielding, maybe the noise is from something else in my circuit. Possible but I do not think it's worth it given the time and effort involved and the thousands of pages of standards documentation and skills that will need to be picked up
But I do not think this research project is worth it.