Whether your hands are neutral, pronated or supinated doesn't actually change anything. What's far more important is grip width and the position of the upper arms / elbows. Having your hands in a neutral position (like on the deadsquat bar) rather than pronated on a standard bar will generally cause you're arms to stay by your sides rather then flare out. If you're arms are close to your sides while you're pressing, then as you press you will be performing shoulder flexion, which will place more emphasis on the front delts and upper chest.
If you're elbows are flared out to the sides, you will be performing more of a horizontal adduction movement, placing more emphasis on the overall chest. As for triceps, the position of the upper arms isn't too critical, as it's more about the angle of the forearms. Having your wrists directly over your elbows is usually a god idea if you don't want your triceps doing too much of the work.
Also, trying to spread the bar apart will help activate the triceps, where as trying to squeeze your hands together will active much more chest. Hope this helps!
While that post is full of fine theoretical pieces of advice. It is not 100% correct or at least it's too technical for nothing. Sure it's not the hand position itself that changes muscle recruitment (although it does play a role) BUT with the DeadSquat bar it DOES put your whole body in a different position when you take the neutral grip. So in that sense "neutral grip changing muscle recruitment" is corrext because it is the action of taking that different grip that makes you put more emphasis on different muscles.
You know I always HATE when people try to sound too smart for no reason. When I give seminars to coaches that's one of the first thing I tell them: stop giving advice to make yourself look smart; give advice to help others out. Now, when you talk to other coaches highly theoretical talk is fine since most of them crave that sort of stuff (the good ones are technique geeks). But for the average person who only want to know what effect they will get, throwing in too much unecessary stuff often clouds the message.
As I always say "Complexity is the langage of the simple mind".
You know, if I answered every question I ever got with every single piece of theory I could throw at the person I would be totally unreadable by 99% of the population and would have likely never bacame a popular coach. That having been said, when I started out training people I DID try to show off my knowledge when talking to clients... and it quickly led to a reputation as a know-ot-all smart Aleck, that was very bad for business.
Furthermore, from experience with doing a lot of pressing with the Dead-Squat bar, and having a lot of athletes press with it, I can tell you that the main effect is a much greater triceps activation as well as more stimulation on the outer portion of the chest. The delts actually received LESS stimulation. Sometimes something that is true in theory doesn't always reflect reality 100%. It's like sex... if you want to learn to satisfy a woman you can ask the virgin who watches porn all day long, or ask the guy who actually has been with a lot of women! While theory is important and give me clues, most of what I learn about the effect of exercises I did by training hundreds of clients from the average Joe to pro bodybuilders, or athletes from 27 different sports.
Also, writing a technical piece of advice is fine within the proper context. For example, when I write an article or a book I do include sections that explain the theory. And that's fine since those who read it are into that sort of stuff. But when someone asks a simple question, most of the time he only wants a simple answer that he can take to the gym right now.
So, if doing 2 pressing days per week, it could actually be a pretty good strategy to have one Dead-Squat pressing day and use a more delt focused pressing variation on the other day? Assuming we're going for max overall development and not trying to correct a specific weakness.
I've noticed that whenever people ask me for advice, it works best when I give them the absolute minimum. For example, if they get a movement straight away, I just point out the basic cues so that the movement "sticks" and to make sure it wasn't a fluke.
If they have a problem with a particular element of the movement, I try to get them to understand through feeling - e.g. understand how elbows move during pressing variations by having them feel out different positions and see where they feel best and how it fits into the movement. They don't have to understand the exact biomechanics, they just need to feel what's going on. Only if they're really interested (e.g. my flatmate, a football player) do I discuss the fine details with them.