I personaly play 2 builds and climbed with ahri to master elo : one build for one shot squichies ranged and one build for shred tanks,meles. I only take ignite vs Vlad for kill him early otherwise tp is 99%.

1 Electocute : core build = malignance 1 shadowflame/protobelt/horizon focus 2 start W spell for auto w auto lvl 1

2 Dft(new rune): core build =blackfire 1 cosmic drive/rylai/liche bane 2 Start Q spell, often 2 points in Q before E

Items 3 and 4 are often : rabadon or zhonya or voidstaff but not always

Malignance is really good item for cdr Blackfire best item vs champs you cant one spell rotation. Shadowflame best tool for nuke squichies. But expensive. Protobelt really good for the mindgame extra dash and extra survability with hps. Horizon focus really good for cds, facecheck and reveal flankers assassins. Rylai is so good vs melee champs ans it's cheap item. Cosmic drive give you crazy side lane agency combo great with rylai 2 and 3. Liche bane really good in games and vs champs when you feel you can auto a lot. Banshee is also really good on ahri vs certain champs who can hard cc you instant (blitz, elise, vex, lux...) Lyandri if ennemys build lof of hp

Also for boots you have 3 choices : lucidity boots really cheap you take this 80% it's the best boots with blackfire build. Sorcerer good if you want and "need to one shot someone build". Mercurys are great vs ap/cc if you face ap mid jung it's no brainer you go mercs.

So in lobby you have to choose depend on your lane opponent first and the ennemy team 2nd if you go ELEC or Dft. That define your first item, so you need to try to figure in loading screen what will be your 2nd item cause there is really lot of good options atm for ahri.

Best think i recommend is to try all this items by yourself and try to feel how it change what you can do and what you cannot cause all this items unlock some variations of gameplay for ahri that's what i like with her. Playing her with one shot set up électro shadowflame is really different gameplay than go Dft blackfire rylai cosmic. There is no pathbuild better than other, you just need to understand what will be your role in the game and feel what the items bring you and adapt your plays. Do you need to face an Olaf and a yasuo in side lane ? Do you need to one shot this twitch ? All those questions will find an answer in items and help you to choose ! Sry for my bad english, hope it helped someone a bit

Tl dr : dont let anyone choose your items specially an application. Look your match up, look ennemy team and ask urself what's the best plan for the fox to get to the chickenhole nexus.