Yes, it sounds like the same thing. Both times when it happened to me the car had only been off for a few minutes. I did try to start it after the tow truck dropped it off at the shop, but even then it had only been maybe an hour and it still wouldn't start.
I'm glad to know that you fixed yours. That tells me that they probably replaced the correct part on my car!
(It was $60.00 for the part, by the way. The part cost included overnight shipping.)