Running CCleaner to clear your browsing history will essentially do the same thing that logging out does - if you’re using its default settings, CCleaner will be removing the cookie in your browser that works as a temporary access token for online banking, but logging out will have removed or invalidated that token already. So no, there is no need to do this, as long as you make sure to log out ️
It’s also totally safe to add the online banking website to your Favourites! Doing this doesn’t save any information that an attacker could use to compromise your account.
For a little extra background, BNZ’s online banking security is pretty good! Although passwords can only be 8 characters long (apparently they’re working on this), your account will be locked after a small number of unsuccessful guesses (I think I read recently in another thread that it was something like 4) so it would be near impossible to crack or guess a strong password. On top of this, the use of two factor authentication (NetGuard) means that even if someone did manage to obtain your password, they would also need physical access to your NetGuard card - for an attack to be remotely possible, the attacker would need to know you personally, ruling out remote attacks, and even phishing, just about entirely.
You’re totally right to be paranoid about online security - there are an awful lot of malicious people out there who would love to get into your bank account. But thankfully, banks are also super paranoid about it! Particularly with NetGuard, you can rest assured that you’re safer logging into online banking than you would be compared to most other websites.