They used to, but no longer ever since 1.6 Weapon/Spell Power normalization.
You want Spell Crit for Staves, and Weapon Crit for everything else. In general, if an ability costs Magicka to activate, it will scale off Spell Power/Crit. If the ability costs Stamina, it will scale off Weapon Damage/Crit.
Dean the Cat
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But, regular attacks don't cost magicka or stamina as far as I know. Also, it makes sense that regular attacks with a staff are considered weapon attacks. Can you please clarify?
In general, if it looks like a spell, it goes off Spell Crit/Spell Damage. Otherwise, it's Weapon Crit/Weapon Damage. Stave light/heavy attacks fall under the catagory of "Looks like a spell", so ZoS made then scale of Spell Damage/Spell Crit.
Staves used to scale off Spell Crit/Weapon Damage back in pre-1.6, but that confused a lot of people, so it got normalised in 1.6.
Dean the Cat
Somewhat Insane Puddicat
EU-PC Megaserver; Ebonheart Pact, Alliance Rank 34 This one hails from far Singapore, excuse this one for his high pings. He also apologizes for any formatting/spelling errors, as he tends to answer using a mobile device.
Insanity is the price of Knowledge. Herma-Mora and Sheogorath, this one bows before thee.
This one does not advocate for any class to be nerfed. There are far deeper underlying issues then a simple "Class Imbalance". The Champion System is the problem. Not classes.