Why not just say "I'm with EMS" as EMR is an EMS provider level.
Or...say, "My name is XYZ, I'm an Emergency Medical Responder and I'm here to help." People who are in a true emergency won't care what it is, only that you are in control and helping them and acting in a professional manner. People who aren't in an emergent situation might say, "What's that?" And now, you have a teachable moment to educate them. Don't go into details about your scope of practice, simply explain that it is the first pre-hospital provider level in the EMS system.
ETA: lying is terrible; lying or misleading someone about your provider level will either result in fines, restriction/removal of certification/license, or possibly much worse. Some people don't know what a paramedic is, but they know what an EMT is. I don't simply say, "yes, I'm a certain type of EMT." I either fix the emergency because they don't care who or what a paramedic is, or if they do care, I just say it is the highest pre-hospital provider level in the EMS system. If they have questions and they have time (non-emergent), I take the time to answer their questions. You are a patient advocate. You are there to make sure they get the best care possible. That includes informing them not only in what you are doing, but also who you are. And, when you share with them honestly, they become more comfortable with you, and this allows for better care.