How is it different if:
a) During character generation, you roll a result that says, "You carry a secret that (insert secret)"
b) During character generation, another player says, "You carry a secret that (insert same secret)"
Is it that we can assume (or we hope) that the player chooses something that fits the evolving fiction, whereas the random table is the same random table for every game in the world and might not make sense for your campaign?
Here's one difference: presumably if someone chooses your secret during PC gen then they are invested in that secret, and so it will matter in play. PbtA games rely quite a bit on this way of building up investments of one player in another's character (eg Hx in Apocalypse World; Bonds in Dungeon World).
On the whole "explore your character" thing I'm still not understanding the difference. Or, at least, why one is "explore" and the other is not. In one case you're making a choice and in the other a choice is being made for you, but if it's leading to something you weren't planning, it's still "exploring" new ideas. Sure, you may choose the well-worn path, but the dice might also choose it for you.
I can see how somebody who never explored new ideas might benefit from the dice (let's say 50% of the time), but somebody who always explore new things would be constrained by the dice (the other 50% of the time).
I'm not sure what sort of dice mechanic you have in mind.
I've already given an example of an actual system upthread: Burning Wheel's Steel checks. The purpose of these is not really to provide new information about my PC: that's a byproduct of using the system in play. The purpose is to create in me - the player - the same sense of apprehension that my PC experiences when confronted with gore or horror, or when trying to do some cold-blooded thing like murder an innkeepr for his strongbox. When I say
I draw Hearseeker - my black-alloy long knife - and stab the innkeeper, and the GM says
make a Steel check, this is not just arid mechanics: this is the game reinforcing the fiction, the reality and enormity of what Aedhros is about to do.
There are other elements of the BW system that are relevant, too. Traits can be voted on or off via periodic "trait votes". And some traits (eg Cool-headed, Cold-hearted, Unfeeling, World Weary, etc) can reduce hesitation in various circumstances, reducing the effect of a failed check. The way that I succeed or fail on Steel checks, and then respond to that in the play of my PC, helps shapes the context for subsequent trait votes. (Eg so far it doesn't seem anyone would suggest that Aedhros should get the Cold-Blooded trait.)
I think this is an example of the sort of thing that
@Ovinomancer is pointing to, which is different from unilaterally authoring one's character and expressing that character in play.
why the cut-off line between physical and mental: it's because it's the only clear, objective line. If that line isn't there, where is it? Is it arbitrary? Since you tend to argue things by showing how it breaks if taken to the extreme (at least, that's the pattern I'm seeing from your posts in this thread) let's imagine a game in which the player doesn't get to make any decisions: you have to roll for every choice. On your turn in combat (which you got into because you looked up the Engage In Combat probability on the table on page 1,417 and rolled less than 32%) you roll on another table to see what your action is for the turn.
You keep saying how this is a roleplaying game and therefore it's about playing a role. Well, what comprises a "role"? Whether or not a sword cuts you? Or what decisions you make and emotions you have? I'll assume you'll agree it's the latter (if I'm wrong I'll be interested to hear the argument). So if you're not deciding emotions and actions, are you actually roleplaying?
Look, I get the arguments for why some imposed emotions/decisions can be interesting roleplaying...it's not my cup of tea, but I can appreciate the appeal...but can't you just lean on that for your arguments? Isn't "fun" a compelling enough reason? What's the point of claiming you don't see the difference between external (physical) states and internal (mental) states?
As far as
playing a role is concerned, at least in the context of a RPG, I don't really feel the force of the physical/mental contrast. What defines the role of a fighter, in classic D&D, is physical prowess (it's not until the cavalier in UA and the samurai and kensai in OA that we see
immunity to fear called out as a feature of the warrior role).
As far as systems where every action declaration requires a roll to locate that within the PC's emotional state, the only one I know of is Wuthering Heights. (The closest to this in more mainstream RPGing is Pendragon with its Trait rules, but typically that's not
every action). Eg here is an extract from Wuthering Heights (links posted not far upthread):
Despair Checks
To make an important decision the Persona should roll above his Despair
To be sincere, a Persona should roll below his Despair
This is also a game in which, as part of PC building, every character must have "a feature floating in the wind (hair / coat / scarf / kilt / whatever)."