Interestingly,, the DOJ and federal courts have no difinition for the term “person of interest”. It is simply a euphemism for “Suspect”,, and it’s catchy sounding for PIOs. It sounds like something an actress on CSI might say.
Senator Grassley even asked the Ashcroft DOJ to define the term in reference to it’s use in the 9/11 anthrax case, and after the response, concluded that it actually HAS no official definition.
Now the Brits,, thats an odd world with some very different approaches!
Between the item one and item two of your list, there's a middle ground best described as “this could be the guy, but we're not sure”. In that circumstance, there has been no arrest, the subject is free to leave, but there is a possibility he's the guy. It's not uncommon under those conditions to continue the investigation looking to compile sufficient evidence pointing at the individual that the level of interest arises to the point where there's a decision to take the subject into custody. In those circumstances, where the police have not yet focussed on a particular suspect, interviews are sometimes conducted in places like hotels, etc. Those are consensual, non-custodial interviews, and no Miranda warning is required. At the point where there's a focus on that individual, or that individual is no longer free to leave, the Miranda requirement is triggered. About ten years ago, San Diego had a kidnap-murder case where a couple of days of hotel interviews were conducted prior to arrest. “Person of interest” would be an appropriate description of the subject at that time.
I agree, it's not a formal status, but in the above circumstances, and in similar situations where a desire to avoid alerting the individual in question to just how much is known and how much trouble he's in (provoking him to flee), it's a useful category.
That said, it makes no sense in the L.A. situation, because the guy's in custody. “Suspect” is the operative phrase there.