Hey Katana, nice to see you here ?
Okay, a lot to work with here haha
First of all, I'm not sure how Vegas works with HDR content as I haven't used the program in years. Based on your output, I'm guessing it likely is outputting in the correct format, the raw PQ/2020 output, which indeed looks "washed out" on an SDR monitor. If you had the ability to output to a monitor that could display that information properly in HDR mode, most likely it would look fine. HDR uses a different "gamma" and color gamut than SDR, so if a program interprets it as if it were SDR data, it looks wrong. That's fine in this case because that's exactly what we need for Voukoder to work properly with the footage. The problem with Premiere is that Premiere would "correct" the display to match the SDR gamma and colors, which meant all that additional HDR information was lost on output to Voukoder. Thankfully, based on your results, I would guess that's not happening with Vegas.
As for Limited/Full, since you're capturing internally on the PS5, I don't think this will matter, because the video files will be encoded with whichever one the PS5 was set to, and Vegas will almost certainly interpret these correctly. So when you output in Voukoder (yes, the output should be set to Limited) then the editor/voukoder correctly converts the input color range to the output. So if the input is Full, then Voukoder will be converting that full range to limited. If the input is limited, then most likely what's happening is the editor will still be interpreting this as Full range floating point RGB, and Voukoder will then correctly map that to Limited range. While you can use Full range, it's not how HDR was designed to be used so I wouldn't recommend it (there's also no reason to use it). As for what to set your PS5 to, it would depend on your TV and how your TV is calibrated. Some TVs work fine with automatic settings, whether on the tv, console or both, but some need to manually adjust this (even if they claim automatic works) and in those cases I recommend using Limited range because other devices will be using limited range and it will just make the TV settings match up correctly. That being said, forcing Limited Range on the PS5 will likely result in less rounding errors or banding in the final output, as values will line up perfectly with the 10bit Limited output, so it might be a good idea to use that.
EDIT: noticed you also have some kind of external capture as well. Assuming this capture device isn't HDMI 2.1, then most likely the PS5 is outputting in YUV422 for HDR, and the output of the PS5 will always be Limited in this mode, even if you have it set to full, and this is good because most external capture devices will expect a "legal range" (limited) signal from YUV outputs, not full range.
As for why your video looks darker on youtube, this is most likely youtube's automatic tonemapping. If you aren't using my HDR metadata tool to analyze frames selected from the full video for the MaxCLL and MaxFALL values, and I don't see that metadata in your MediaInfo outputs so I'd guess you're not, then youtube has to make assumptions about the HDR content of the video, and try to map that content into SDR without severe clipping or crushing of highlight information. I personally use a process to create my own LUT which totally controls the color grading and tonemapping process for the full video, but that's a pretty complicated process as well and if you're not familiar with color grading and LUT creation, it's better to determine the correct MaxCLL and MaxFALL values and let youtube handle those. It may still end up looking dimmer than what you'd expect because HDR in general tends to be brighter than SDR in many cases, and youtube will need to darken that to avoid looking overexposed, but as long as it's presentable I wouldn't worry too much about that.