Talking objectively here... I realize that each one has different benefits, but which one will net you the most income? I feel like it's Relic Shield, but I haven't actually done the math or paid enough attention to the items to know for sure.
um Well I used to Buy SpellTheifs On gp But Then They Decided to Kill the item
It really depends on the champion. A poke-heavy support could keep Spellthief's constantly on cooldown, meaning that it's earning them a ton of gold. With a melee champion in a lane where the enemy bot lane can't deny him very much, Relic Shield will give him a ton of gold by allowing him to share his ADC's cs. Ancient Coin has less consistency and player control over the item, but it doesn't require you to actively use it, making it the easiest to generate the most gold on of them all.
I find it's coin.
I'll win lane hard with Lux sup (for example) and the gold quest will complete 4 minutes late.
With relic, your team doesn't give a rats about letting you on the valuable CS a lot of the time (complete the quest later then, too).
I can't speak for 'perfect' scenarios for the gold income, they really don't happen.
Depends on who ur playing. If u playing a heal/shield champ then prob ardent.
Depends on many factors. High elo coin is the defacto worst option, but i find spellthiefs can be punished the hardest (minion lockout and if you fall behind you cant push or harass safely to earn the gold you need)
with the massive increase in lane sustain and changes to quests, i have started to ignore spell-thief and go for coin, simply on the account that there is less hassle attached to it.
I know it's a bronze move, but honestly the whole quests and gold generation mechanism things that these items have, the whole thing combined and each component individually, is a sucky experience for me.
This item is most useful in late game among the other support items. It gives movement speed and CDR which is very important late game also it gives money over time + money from minions. Having this item even if your lane is pushed hard, you can still have more income, because tower kill minions and you get gold. While with other items it is more complicated.
The downside - when you roam and hide in the bush, enemy can spot you when they see a coin drop from dead creeps..
Now many people start with relic and go 
although the active of this item is pretty good but in early game if your adc cant do much you are also put on disadvantage, less income. You can execute minions healing your adc but this cant really be used in the middle of a fight when this ability is the most needed.
Before the Ardent Censer nerfs, people used to cry a lot because of how easy it was for Janna to get gold enough to buy Ardent Censer just by having coin.
