Forrest,
Sounds like Haley was made for that role, given she grew up with an architect father! I can imagine she aced that audition......
While I haven't played there yet, Stream Song is the obvious place to make comparisons of those three styles. You can learn some from the web site photos.
As to distinctions, in the 80's there was probably Pete Dye (who you left off the list) and everyone else. Without railroad ties and long strip bunkers, most of the rest were pretty much the same, even to me. JN had the steep banks, which stood out for a while. Maybe copied from Pete, but always cleaner edge.
Otherwise, the traditional RTJ/Wilson style carried on for most architects in that era, modified only slightly by each practitioner. The differences would be interesting for some grad student to study. Dick Nugent said he could always tell the designs of the different associates in his office, and declared the difference to be "wrist radius", or how broad the free form lines were drawn, not out of any conscious decision, but because that's just how long the arm was.
If you could tell a Faz, it was only because the details, landscaping, etc. were all done up more than others.
IMHO, Hills and Palmer had the rep of churning out housing courses of similar styles, and to a degree (obviously each had a few stout successes) they deserved it, because they chose to tackle the housing market, and design for the particular needs of that genre. Nothing wrong with that. Most architects over time take as many projects as they can, and consider 1-3 to be "lifetime" opportunities.
Of course, that looks backwards, which is always easier. As to the current group.....hate to comment without full benefit of hindsight, and having played a limited sample size, but have some random observations.
I saw High Point, and realized TD was really trying to do something different than the rest, so that always stuck out in my mind. And, perhaps as a result of that, I still think his style is the best among those three. Conceptually, really trying to use the land is "purer" than trying to copy the early American designers,(CC) or mimicking your mentor, but with enough twists to make it your own style (GH) and I think that underlying design philosophy shows up in the finished product.
At places like TT, TD mimicked the bunker style of MacKenzie, which were different in form (mostly skinnier) than a typical RTJ/Wilson copy bunker. I associate CC with rough edged "sweet potato" shaped bunkers, as opposed to cape and bay style of RTJ (or even Mac). However, when I saw TD's Hainan course, I noted he used simpler shapes, albeit, with jagged edges. (I saw it with only six holes shaped) Can't say if it is a trend or a one off........
Both TD and CC seem to be great at randomness. Comparing Sand Hills to JN's nearby Dismal River course, you can see Nicklaus tries to put more "organization" in his design, and his holes fit the topo more like a typical course would fit man made mounds, i.e.. going around them with some sense of design purpose, which doesn't work quite as well as CC style at Sand Hills. Have heard some in the biz say CC was probably the only architect who could have gotten those results at SH, and I agree. Most others probably would have thought more traditionally, and ended up with something similar to Torrey Pines - a standard course in an exceptional setting.
Haven't played a Hanse, but from pictures, Gil's style leans more to the JN organized style and is the least rugged of the three, and in my mind, ruggedness is one of the hall marks of the current style. That said, Pete went from rugged to refined over the years, and probably all three of the current guys will end up with projects going that way, too. After all, most high end jobs end up getting high end maintenance, so it sort of follows. Not every project can be Bandon or Streamsong. Will be interesting to see how they adapt their style, or clients adapt their maintenance to a more typical in town country club........and Gil may end up being the architect of choice for those.
As to strategy, well, all three pay homage to the past masters, which would seem to limit differences between them. As always, must MHO. And, these were typed quickly, with emphasis on the word random......call back tomorrow, and I may have more or different thoughts.