Yes, same thing. "Authorized Representative" is MDD terminology.
You maintain the official complaint files and associated systems. You probably will want to provide your own contact information on the labeling, so that complaints and other communications come directly to you. In this day of cheap international phone calls and essentially free email, there's no reason to direct end users to send communications to your US Agent. It just slows down the communications cycle. The requirement for a US Agent is mostly a legacy formality, at least in regard to well-established device makers.
The agent is declared as a part of your registration...not registered on their own, unless they have an addition role such as Initial Importer.
You do have to have at least one Initial Importer, but that party does not have to take physical possession of the goods or have a facility that is capable of warehousing and inventory. The regulation saying "possession" in my understanding refers to legal possession, i.e. title. Instead you can drop-ship directly to your distributors, as long as the paperwork reflects that legally you are selling to the Initial Importer and they are selling to the distributor(s).
The Initial Importer's role is one of legal responsibility in case FDA has a major problem with you, because FDA cannot impose major penalties on outside-USA companies, other than to bar their products from the US market.
You can have more than one Initial Importer.