Neither term makes any sense. Apple never says "factory unlocked" as there is no such thing. Most people who use that term mean that the phone was hacked in the "factory" behind their store, or in their garage. A phone's lock status is set when the phone is first activated, based on a master database of all iPhone serial numbers maintained by Apple, but populated by carriers.
I've never heard the term "open line".
If you want a legitmately unlocked iPhone there is only one way to get one new: Purchase it directly from Apple, pay full price, and be sure to specify "unlocked". Some carriers will also unlock a phone, either for a fee, or if the original owner's contract has been satisfied for no charge.
If you are in the UK Carphone Warehouse sells phones that it says are "SIM-Free". DO NOT BUY ONE OF THESE. SIM-free means the phone can be used with any carrier, but the first time they are activated they become locked forever to the carrier they were activated with.