The Hocus and the TouchStone. Skinner and Maestro'D

Hocus leads a cult, of course. Still, he is very much "preaching" in my vision - same as Touchstone. He is more "solitary" preacher, yes, more warrior-like with some moves, but I can't get this precise feel, why they're crucially different. Why one can't be the other.

Maestro'D owns an establishment, of course. Why Skinner can't? (I can go with same ideas as above, but I suppose you've got it)

They're very similar in feel. Can somebody please describe their critical differences? With examples, if possible.

I see the Skinner actually being sort of polar to the Brainer. Both are about controlling people but the Brainer controls people in a weird direct way and the Skinner in a hot indirect way.

The Maestro'D is not about controlling people at all, it's about information and networks. Everything passes the establishment at some point and the Maestro'Ds moves are about catching them.

Sure the Skinner can have an establishment as well but it will be about people going there and falling in love with the Skinner and its art not gossiping and fraternizing.

The Hocus' cult can be about whatever. The Touchstone is preaching about clean water or something. The Hocus' is very close to the maelstrom and can sometimes use its followers to manipulate it. The Touchstone is very grounded in the old reality without maelstrom (-1 weird on all 4 rows if I remember correctly) and fighting battles and having a long life feels very much like a real person with an idea as opposed to some crazy as vessel for a higher destiny.

Okay, slightly easier with TouchStone. So basically he's more "sane" and "old-normal-ways" preacher(though in his book there's definite mention of "new ways"), while Hocus is crazy hand-waving guy, who can actually do some weird shit.
Normal against Weird

Still no good with Skinner. Is it that everyone just should fall in love with him/her, or there's something beyond that?
In Firefly-'verse Skinner vs Maestro'D is Inara vs Nandi?

Having played the Skinner and Maestro both, here's what I experience:

The Skinner has a lot of ability to create opportunities. He controls peoples' attention and has the ability to influence them from afar. What the Skinner fails at is being able to capitalize on those opportunities under his own power - he needs to control people to be effective.

The Maestro has a lot of responsibilities to the establishment. When you're the Maestro, most of your time is spent solving problems while you're trying to throw a party, or alternatively it's spent using your Big Knife to try and get you out of trouble. The Maestro has some basic ability to control people but his most effective moves are about managing resources.

That said, I think it is best to see the playbooks as starting points for characters. Since you can change playbooks often, and even prior to that you can change your stat balance and/or take moves from other playbooks (my Maestro has Lost and Reality's Bleeding Edge because she's getting really Weird), the playbook itself isn't really much more than an abbreviated chargen guide. There's no specific reason that you can't have an establishment and also Skinner moves, you just gotta make it that way with improvements.

On the Maestro D' and the Skinner: the Maestro D' makes an excellent "prestige class" for a Skinner who's become an "ascended performer" and obtained their own establishment. I was actually leaning towards that with my own old Skinner before the campaign got canned (sadness).

(I will admit that the class-changing rules as written can be, uh, problematic, though. It basically means that moving from a class with a lot of intrinsics to one with a lot of extrinsics is awesome, but that if you're a class with a lot of extrinsics and you try to pick up one with intrinsics, suddenly you're screwed and damn near starting over.)

Also, a cult leader and a warrior fighting for a better tomorrow are two entirely different things. They can merge, yes, much as any two archetypes can--the Touchstone actually does have an improvement that gives them followers and the fortunes move. Think of it this way: the Hocus is not only supernaturally oriented, but a leader (and they may even be one more interested in power and exploiting their cult than their stated goals), whereas the Touchstone is more naturally oriented (as observed), and while they absolutely can lead and especially inspire, they do it from the front lines: they are hero first and leader second, and all that. Though you seem like you might already have that down, so I'll  reiterate they absolutely can become the same thing: it's just a matter of things like getting cross-class moves, finding ways to bolster a Hocus's Hard and a Touchstone's Weird, and probably going on to change playbooks when one gets the opportunity.

Related to the idea that the Skinner and Brainer are polar relatives is this custom move the Skinner took in my game:

Pillow Talk: When you spend physically intimate time with someone, you can get them to tell you what makes them beautiful, strong, or worthwhile. Roll +hot. On a 10+, hold 3. On a 7-9, hold 1. Spend your hold to ask their player questions, 1 for 1.

- What was your character's proudest moment?
- For what is your character most thankful, and why?
- What are your character's guilty pleasures?
- What is your character's source of strength?

On a miss, they get to ask you a question of their choice instead (not necessarily on this list).

the Maestro D' makes an excellent "prestige class" for a Skinner who's become an "ascended performer" and obtained their own establishment.

Totally. This is verbatim what Kat did in our Appalachia, Treehouses, Windmills game – took October the Skinner (who ran a place of ill repute the Moulin Noir that happened to serve tea) and made her October the Maestro D (who runs a teahouse offering "extra services").  And the shift is noticeable:  now she's more concerned about her establishment/employees than about indirectly manipulating the other PCs.

Hocus leads a cult, of course. Still, he is very much "preaching" in my vision - same as Touchstone. He is more "solitary" preacher, yes, more warrior-like with some moves, but I can't get this precise feel, why they're crucially different. Why one can't be the other.

The Touchstone can be a preacher, but they don't have to be. The key element of a Touchstone is that they have hope, and give people hope (otherwise their abilities don't work). Anybody can do that, they don't have to carry a book. I'm not sure I agree that the Touchstone is grounded in the past -- by definition, they have a vision of the future, and presumably hope for and work towards it.  An important distinction between the Hocus and the Touchstone is that everybody the Touchstone meets will give them food and shelter (which is a big deal in Apocalypse World), whereas the Hocus is only special in their little cult.

It's also a big deal that the Touchstone nominally works through violence. It's at least looking up at you during the game.