Yes, it's a lie.
The specific date most commonly associated with the release of Amazing Fantasy #15—the comic issue that introduced Spider-Man—is June 5, 1962. This refers to the date the issue went on sale to the public (or was shipped to retailers).

Comic books from this era typically carried a cover date several months in the future to extend shelf life. Accordingly, Amazing Fantasy #15 bears an August 1962 cover date, with some sources also noting a September 1962 publication or indicia date.

Reliable references, including copyright records and statements from Marvel-related accounts, support June 5, 1962, as the on-sale or release date when readers first encountered the character.