I see your dilemma. This is why kids cannot and should not make their own social arrangements until a certain age. Literally, this is "Playing Telephone" - you don't know what the other mom said, or what the other kid said. You only know what your kid told you. Just one or two details needs to change from one party to the next. So the mom could have said "meet us there" and the friend said "come with me", or he could have said it right and your son "translated" it incorrectly.
The other kid's parent should have said, "Ask Jimmy if he wants to meet you at the movie theater, and tell him it costs $10." Let the kids "practice" making a date (social skills development) and then let the parents get on the phone to verify/solidify the specifics. That's when the other parent could have said, "Can you drop off at 3:00, we'll meet him at the entrances, and I'll bring him home since I have to do XYZ first?"
Or, you could have gotten on the phone with the mom after the boys made the first attempts, just to verify. So, while I see why you assumed what you did, it's a lesson learned that we just can't always do that with our kids.