There's quite a lot to unpack in this, I'm afraid. Firstly, no you can't do what you want - well not the way you want to do it, anyway. Why?
In Waveform view it's only ever been possible to record two channels at once, and that's because it doesn't record direct to file, but to a temp file which only gets turned into a recorded file when you save it. Back when this was instituted, it had enough trouble even doing that... and also, Microsoft made it difficult to use multichannel drivers by charging a fortune to have them approved to work with their OS's. So nobody went down that route, and recording in that view remains resolutely two channel only. Waveform view does have a trick up its sleeve though, which we will return to.
So Multitrack view was added, and eventually that got a record facility that was Multichannel (If you use ASIO) and direct to file. But as you've found, it doesn't quite do what you want because the mixdown seems to be rather inflexible. So what do you do? Okay, it's a bit of a kludge, but it does work...
You create a new session with a separate track for each input so that you can record simultaneously - it doesn't actually matter about how it mixes down, because what you are going to do is use the Export facility, and create a separate file for each of your input tracks. So then you've got four separate files. Now we get around to Waveform view's little trick - it can create Multichannel files! Go to File>New>Audio File and select Custom. You get a screen like this:
Where you can create whatever file you want as a container. All you have to do now is copy each of your exported files to the appropriate channel in it, and you're good to go. Yes you might think it's a bit of a pain, but I'm afraid it's the only way in Audition to achieve what you want. And it is very flexible - I've used it for B-format files for years.
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