That pretty much covers it. An EMT (Emergency Medical Technician) is a person that has completed the proper training to receive the title of being an EMT. EMS (Emergency Medical Services) is a system of people and equipment that includes but is not limited to EMT's, MICN's (or RN's that do pre-hospital care), ambulances, helicopters, etc. Usually these services are activated by the 911 system and they provide emergency response for sudden illnesses or injuries.
An EMT-B certification is a 120 hour class and a state test following to be registered. A EMT-B is considered to give care that is at the BLS (basic life support) level.
An EMT-P is much longer, an EMT-P is kind of a pre-hospital nurse. They can start IV's, push certain drugs, provide ACLS (advanced cardiac life support), and do a more detailed assessment.
An EMT-I is kind of an in between. (I wish I knew more about their guidelines and what they can do, but NJ does not have an EMT-I level)
So hopefully between these two responses you can understand a little more of what both abbreviations mean and stand for.