The bounty system got changed some time ago, and now bounties persist for a fixed time duration depending on the severity of the bounty. Once the timer has expired, it switches to a 'Dormant bounty' meaning you won't be broadcasted as wanted but if a ship scans you with a K-Warrant scanner it will trigger the dormant bounty and make it active again.
So if you randomly attack (But don't kill) a clean ship and get a 600 credit bounty it may have a cool off period of 10 minutes or so, during which you'll be broadcast to everyone as wanted. Once those 10 minutes have gone by, it will switch to a dormant bounty with the same 10 minutes duration. Whilst it's dormant you'll not be shown as wanted, but if you get scanned by any ship using a K-Warrant scanner the bounty will reactive and the original timer on it will reapply so you'll be back to being wanted with a 10 minutes cooldown on your active bounty.
If you avoid being scanned until the dormant bounty timer has ticked down, then it will transition to a legacy fine which you can then pay off at the local security office.
Keep in mind that committing new crimes in a system where you already have a active or dormant bounty will add on to existing bounties or reactive dormant bounties and add the new amount to the existing one, and the larger the bounty the longer the cooldown timer becomes up to a maximum of (I think) 7 days. Meaning if you have a 7 day bounty, it will actually takes 14 days before you can pay it off as a legacy fine... 7 days for it to transition to a dormant bounty, then that dormant bounty also has a 7 day cool off period. If you're scanned in the system before the last 7 days are over, you're back to a 14 day cooldown.
Whilst I haven't tried it myself, I don't think that being scanned by a K-Warrant scanner in the system you don't have the dormant bounty with will reactivate it, instead dormant bounties are only reactivated if you're scanned in the system they're held in. But I wouldn't take that as fact.
