I found these separate pouch underwear from a brand called "David Archy" on Amazon; I never could find an official website. Regardless of where exactly they come from, they are super cheap, really nice material and quality, and very comfortable.
Here's the style that I purchased:
http://www.amazon.com/David-Archy-Separate-Pouches-Trunks/dp/B00DYNPFUU/ref=sr_1_2/187-0929915-8293719?s=apparel&ie=UTF8&qid=1450540198&sr=1-2&nodeID=7141123011. They have a pouch made of thin material for the balls (which I'm sure is good for guys with biological scrotum to keep cool). Unlike the ones that you posted, these have a "pocket" that the shaft tucks into that is attached at the top, with a fly on the underside.
I'm not very good at using my STP, so I don't really use it to pee. However, it packs in these really well (I have the Model D). It is easy to pull the shaft out from the "pocket" if you were going to pee.
They do run small and are only available up to an XL. I've got about a 40 waist, 44 hips, and they fit, but they're a little snug. Also, I think that the care instructions suggested that you hand wash them. I was them in cold water on delicate in a garment bag, and hang them up to dry.
Overall, for 25 bucks for 4 pairs, I would give them a try.