[there is a longer tl;dr at the bottom]
The short, SHORT answer is OBS, for everything :)
I'm surprised he was streaming with something else. It can be a wicked complicated bit of software, but the defaults are all sane, and it's very user-friendly when you decide that you actually want to record B-frames ever 2 instead of 4 seconds, or whatever else weird corner you end up being very concerned over :) Streaming & recording at the same time is basically what inspired the program in the first place, hence the name.
Frankly, I wouldn't touch anything else, because I don't know what it's doing under the hood. Even if I don't change anything, it's important to know what it's doing, and how.