i think the real point of distinction becomes what to do in mid to late game. everybody should be pitching in to the citadel until ALL of the following upgrades are purchased: building health 1, building damage 1, gold income 1, experience 1, creep damage 1, creep armor 1.
once those are bought then you get to worry about who saves up for artifacts vs. who saves up for catapults.
each Demigod has a different method of farming and pushing so they should trend towards buying the upgrades that best enhance their farming method.
Beast farms almost exclusively by hunting enemy DG's. he's noticeably poor at creeping. Pure damage works well for Beast. he's a good carrier for Mageslayer and Ashkandor.
Regulus can creep extremely well with Angelic Fury and Mines or he can hunt demigods extremely well with Snipe and Mark of the Betrayer. its very hard to play a build that does BOTH well. Angelic Fury build can also be deadly to demigods but probably not before a significant amount of attack speed gear has been purchased. If you're mainly farming and Pushing with Angelic Fury you probably don't need much gear just for that so you can buy alot of Citadel Stuff. if you're hunting Demigods (regardless of your build) you're almost definitely going for the best attack speed gear you can find so Mageslayer, Cloak of Flames, and Ashkandor are on the top of the list.
Torchbearer is noteworthy in being able to play without much equipment. it doesn't really matter what your build is as long as you've got a Blade of the Serpent, which will provide you with as much mana as you ever really need. Torch is extremely good at creep farming as well. this suggests he should buy the creep and experience upgrades at the citadel.
Rook is an extremely good pusher and creep farmer. he's bad at killing Demigods unless they try to confront him in a tower farm, which they won't if they're smart. Tower builds of rook are also noteworthy in being almost totally indepent of gear. Rook is also a good choice to buy alot of the creep and experience upgrades at the citadel.
Sedna is great at killing enemy Demigods and great at capping flags. she's very bad at creep farming and only barely average at tower pushing. her mana use is extreme so she needs to buy alot of equipment. i think for this reason Sedna is a good choice to save for artifacts. I'm quite fond of Orb of Veiled Storms and Bulwark of Ages on Sedna.
Queen of Thorns is massively strong at pushing and creeping but she chews through mana just as fast as Sedna. you can easily spend 10k gold on mana items as Queen before you're ready to buy much of anything else. Queen is a good choice to save for Artifacts as she benefits alot from stuff like Cloak of Flames and Stormbringer.
Oak is very versatile and either of his major playstyles (Raise Dead oriented or Penitence oriented) are good at lane pushing due to Surge of Faith and Divine Justice. His Raise Dead build is not good at killing enemy DG's until the lategame (when a few towers are down and the minions can pursue much further without dying). The Penitence build is vey good at killing enemy DG's and can farm money quickly this way. Either build is only medium equipment dependent. a few health and mana items help alot as do Idols but Oak has a very low need for artifacts. for this reason I think Oak should generally be trying to buy citadel upgrades once he's got the few items he really does need.
Erebus is very bad at creeping and tower pushing. he's extremely good at killing Demigods. this makes him alot like Sedna and thus a good choice to save for artifacts. auto-attack damage boosters like Mageslayer are particularly good for Erebus. Erebus can also be mana intensive (Bat Swarm and Charm both cost alot) so items like Cloak of Flames are good for him as well.