Is there any way we can watch the recording of the guys talking over the Sony livestream?

Self explanitary

Twitch has it archived.

https://www.twitch.tv/giantbomb/v/88108240

Uh, isn't this what you want?

http://www.giantbomb.com/videos/we-talk-over-the-playstation-4-pro-reveal/2300-11544/