- As a note, though the Rules of the Game articles are generally higher-quality than the FAQ, they are no more official primary rules sources than is the FAQ, and everything wrong with the FAQ is true also of the Rules of the Game. They have no place in a RAW discussion. So I suggest you ignore it if you are interested in RAW. That said, the quoted portion appears to be accurate.
Dice-based bonuses are no different from any other bonuses in terms of stacking. They just have a variable component. Thus, they stack if they would otherwise stack.
Most importantly, dice is not a type of bonus. For that matter, neither is cold or fire (those are types of damage). The Modifier Types List is not necessarily exhaustive (particularly considering possibilities present in later publications), but the lack of these alleged types is telling, since bonuses expressed as dice are prevalent in the core rules.
Damage bonuses, especially dice-based ones, are frequently untyped; every case you have listed here is. That means that they stack unless they come from the same source.
So if you activate Psionic Weapon twice (it is possible to get multiple psionic focuses), it wouldn’t stack with itself (though you would be able to roll twice and keep the higher, since in cases where bonuses don’t stack, you still have both).
But Psionic Weapon will stack with Deep Crystal, because the source of the two bonuses of 2d6 damage each are different: the Psionic Weapon feat and the Deep Crystal special material. However, pumping more power points into Deep Crystal will only create more non-stacking instances of 2d6 bonus damage, since each instance of 2 power points gives a bonus of 2d6 damage from the same source, the Deep Crystal. (Compare to other effects that say something like “For every 2 power points,” implying one bonus whose size depends on the number of points put in.)
The 1d6 fire damage from flaming stacks with the 1d6 cold damage from frost because those are both untyped bonuses to damage. On the other hand, a hypothetical flaming flaming weapon would not stack, though again you’d get to roll 1d6 twice and keep the higher. More relevantly, a flaming-burst flaming weapon would also only get 1d6 rolled twice (on a non-critical hit), because “A flaming burst weapon functions as a flaming weapon,” i.e. the source of the 1d6 fire damage on a non-critical hit is from a built-in flaming property. However, if some other property added fire damage to the weapon, and did not build flaming into itself, then that would stack with flaming – fire is not a type of bonus, it’s a type of damage, and the type of the bonus here is untyped.
Finally, compare Sneak Attack: every1 instance of Sneak Attack in the game includes the following blurb:
If an assassin gets a sneak attack bonus from another source the bonuses on damage stack.
Sneak Attack’s damage bonus is still not typed, but if you had Sneak Attack from multiple places, that would run afoul of the same-source rule. So Sneak Attack has this specific rule allowing it to stack anyway, so that you can multiclass between Sneak Attacking classes without having to start your Sneak Attack progression over.
- Technically, the rogue itself lacks this line, which is irrelevant since everything else has it and thus will stack with the rogue’s. The only other exception here is the factotum, which is a weird case where it not stacking appears to be a critical part of how the class is intended to function, but the factotum really should have gotten some more editorial love and much about its rule text is awkward or ambiguous.
Best Answer
Luck bonuses do not stack with each other
From the SRD:
However, bonuses of the same type, but to different scores, coexist. For example, if all three of those spells were affecting you as cast by a ninth level caster, you would have: