Generally ICICI forces to opens 3-in-1 account (Savings + Demat).
If you have wish to open the salary account then bargain with them for LTF Demat account. When I was having salary account with ICICI, I got the ICICIDirect as an free (no account charges at all. Even includes free brokerage of Rs.2500 that time for 1 year).
You can close the Demat account if you don't want it later.
Now depends on salary & investments what type of account you get. If you can invest good amount (or good salary figures) then try for private or wealth.
For no salary account or with less investment but have good credit card limit (>=4L):
I suggest to open the 2 accounts (you can ask your family member) and enroll for Family 360 before 1st July (else debit card annual charges be there if apply on/after 1st July).
For single account, you can opt privilege still --> I don't think if you visit branch and they deny to open the account (they get commission with account opening so you're the boss).
Note:
Family 360 is just addon over Privilege kind of accounts. You can't opt for Family 360 code over wealth/private account as those are high variant accounts.
If you don't want branch headache then give the online account opening try
https://campaigns.icicibank.com/cas...avings-account-main_open-saving-accountCTAbtn