I agree with them just being an asshole. A lawyer is someone with a JD. An attorney is someone licensed to practice law. For a lay person, I don't think it's at all inappropriate to conflate the two, and giving them crap about it is definitely an asshole move.
Fwiw, as married to someone awaiting character and fitness confirmation from 2nd department of New York State - you cannot declare or imply that you are a lawyer, attorney or otherwise able to practice law until duly licensed and sworn in. While Websters says a lawyer includes those that study law, putting that in your email footer is a great way to never be barred.
I think actually it does: the original meaning of "attorney" was someone who was appointed to act for you or on your behalf. An attorney at law does that when you need legal representation. Someone with a power of attorney does that in other areas, such as finances.