I just answered this on another sight and I don't know if I'm about to get a blast for necro-ing this discussion, but it's the top result on a Google search of the subject so I'm going to do it anyway. My answer (confidant theory) is this:
Tackling Bug v Dark:
A key thing to note is that the Japanese name for the "Dark" type is more closely translated to "Evil" or "Sinister." This is why dark types have many underhanded or villainous move names like "Quash," "Sucker Punch," "Fake Tears," "Torment," and "Nasty Plot." Also why the Fairy type, which embodies goodness or purity, has a type advantage.
Alternatively, Fighting type embodies heroism or discipline. If you consider the culture of classic Japanese action films, the heroes who take down Evil are usually adept and disciplined in martial arts.
On the Bug side, insects obviously aren't know to be the undoing of evil, but if you consider what things people (particularly the Japanese) associate with bugs, things become more clear:
- From one angle, we can again refer to Japanese culture. To quote TvTropes.org: "Japan loves bugs, and they're everywhere. Not only is Bug Catching one of the oldest pastimes in the country, but much of their pop culture includes or is influenced by them." Beetles and bugs are regularly the costume theme designs of fictional heroes or mechs in Japanese series. In Gen II, they introduced Heracross a Bug/FIGHTING hercules beetle, a wildly popular beetle, alongside the Dark type to perfectly counter the new type. This is the "Hero" angle cropping up again.
- The second (my preferred) angle to consider is on the other end of the spectrum of how Evil is vanquished. Bugs are known to swarm or cooperate in numbers that overwhelm the target (think endless mosquitos on a camping trip or a synchronized colony of ants). Similarly, if one wants to overthrow a great evil, an army or a cooperative society can collaborate to take it down with sheer numbers. Evil may strike one or a handful down, but the movement will overwhelm eventually.
So ultimately, Evil (Dark) is framed as being bested by discipline/heroism (Fighting), the collective effort of the masses (Bug), or pure goodness (Fairy).
This whole relationship IS much more conceptual that of Ice v Grass or Steel v Fire, but Evil isn't exactly an element.
Tackling Bug v Psychic:
Basically what everyone else has said: phobia that disrupts mental harmony or focus. You should stop reading here if all you care about is bugs.
BUT you may be saying, "Hey now, that flies in the face of the meaning of Dark as you laid out earlier! Evil isn't a phobia!"
Following YOUR tangent, shame on you, I would argue that Dark, introduced in Gen II, was not built in reference to Psychic's weakness theme. Instead, Psychic was made to fit more in line with Dark's theme by creating trifecta similar to Grass-Fire-Water using Psychic's established relationship with Fighting.
Psychic has always had a "mind over matter" kind advantage against Fighting, and Fighting is, as explained above, a disciplined righteous counter to Dark's evil or trickery.
Dark can be seen as the Scar to Psychic's Mufasa. Sure, Mufasa was wise, but Scar was a tricky and underhanded - enabling him to (spoilers) kill his brother.
If you're into Shonen manga/anime consider this trope relationship:
- the young martial arts hero (Fighting) trains under a wise and tactical sensei (Psychic) and can never outthink and defeat him
- the sinister villain (Dark) underhandedly counters the sensei's knowledge (Psychic), defeats him, and reigns supreme
- our hero (Fighting) gets tripped up a bit by the tricks of the villain (Dark) but ultimately, through sheer willpower, avenges the sensei to punch the villain right in the face
So yes, Bug beats Psychic for phobic disruption of the mind, but Dark (as fun as fear of the Dark is) doesn't beat Psychic for the same reasons, as was proposed previously. Don't think of Bug and Dark in relation to Psychic, but instead Bug and Psychic in relation to Dark, the younger typing.