Just to clarify, this thing about logging into the official Runecape site is a myth.
Insta-bans can occur mainly because you have a flagged IP and each account you create on that IP will be banned instantly due to your botting history on that IP, can you avoid this you ask? Yes, you should use Proxies ALWAYS when botting, it's a solid investment and very cheap.
The insta-DC ban when you log in is due to Jagex having an offfence against you, the way bans work is that you are banned when you log in once jagex has put the offence on your account, the reason for this maybe taking "a few days" to place the offence on your account is their way of confusing you and bot developers how they detect botters. It is a "Delayed-ban" as you would say now adays...
Hope that clears this all up ^_^