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The exact formula isn't known because noone bothered yet to make an universally applicable one. Jouni's is the one for non scaling and former pure physical weapons.
1) AR bonus of your stats affects total damage but it doesnt affect the buff
2) Weapons infused with any kind of magic benefit more from buffing
3) You can infuse your weapon with any magic you like, it does not have to be lightning (however, lightning infusion is better for fth builds due to scaling reasons)
1) Overall, the best element in PvE is Lightning.
2) Overall, the best element in PvP is Dark.
Reason: Most enemies' defenses for that specific element are lower than for the rest.
Sunlightblade's duration scales with Faith. Therefore, you obviously want to level Faith. What are the infusions that scale with Faith? Fire and Lightning (and Dark to a degree). Lightning has the highest AR at base Int and 54 Faith (which is the max SLB duration). Lightning gets a tiny "fitting infusion + buff"-bonus.
Numberwise and theoretically, assuming all elemental defenses are the same, Lightning would win. And for PvE it also does in practice because it is the most common weakness and most resistant mobs fall in like 1 or 2 attacks anyway so it doesn't matter.
In PvP however, you'll find opponents stacking defenses when they see weapon-buffs. They can get up to 42% defense to an element by just equipping 2 rings and consuming one item (which lasts for 90 secs so about as long as your buff), that's WITHOUT the innate 10% hidden defense every player has by default and WITHOUT any armor, so you can figure out that you can stack some more defense by wearing appropriate armor. Actually, almost noone will do this, but it should be mentioned.
But when you mismatch infusion and buff, you'll almost completely counter their strategy and Dark is the most common PvP weakness, so a Dark Pickaxe plus Sunlightblade would be worth a thought.
The main reason: The up to 30ish additional elemental AR Lightning has over i.e. Dark from the scaling won't be much of a difference in actual practice. Infact, this whole comment is just about minmaxing. It REALLY doesn't make much of a difference what exact infusion you pick as long as you DO pick one, and it has to be elemental (so Fire, Dark, Magic or Lightning).
Actually, there's one thing which'll skyrocket your damage and that's the Old Leo Ring. For some reason, elemental damage is affected by it and it multiplies with existing counter-damage. And ALL of the Pickaxe's attacks are thrust-attacks.