So what did you get from this part?
'And ‘note that problem solving, in this context, has to do with a problem to be solved by the character, not a problem (such as How do I role-play this situation?) to be solved by the player.'
My reading of this is the same as [MENTION=6787503]Hriston[/MENTION]'s - the player is expected to insert him-/herself into the ingame situation, and then engage that situation. So the player does not confront a problem of "how do I roleplay" - because that has already been answered by inserting him-/herself into the situation. But the situation is a problem that has to be solved.
What I would add to Gygax, in explicating my preferred RPGing approach, is that
if you want the personality/character of the PC to emerge in play,
then the resources available to the player, and the permissible moves for the player, should be designed so as to help bring this about.
One of the clearest and simplest examples I know of this, from 4e: paladins have an at-will ability called Valiant Strike, which grants a bonus to hit equal to the number of adjacent enemies. So when the player of the paladin inserts him-/herself into a fictional situation where enemies are present, s/he has a good reason to want to be in the thick of things (surrounded by enemies, and therefore getting a bonus to hit) rather than skirmishing on the edges. And this, in turn, will make the PC be
valiant - because in the thick of things rather than a skulking skirmisher.
There are some hints of this sort of mechanical/resource approach to establishing characterisation in classic D&D: for instance, fighters get an XP bonus from high STR, so the player of a fighter has an incentive to make STR high, and this in turn makes melee an attractive problem-solving option, which in turn makes it more likely than for other PC types that the player of the fighter will find him-/herself engaging in melee rather than being a skulking skirmisher.
But some more contemporary games (including 4e, in the D&D stable of game) take this sort of idea further.
Needless to say, I think that 2nd ed AD&D is the furthest from this sort of approach to the relationship between desire for characterisation and suites of mechanical options that are available to the player who has inserted him-/herself into the ingame situation.