We were discussing the legendary rat pic with friends. One of them pointed me to the page that was most probably the source of that photo: https://theagilerat.com/?id=36
Of course i had to download it and check if it had EXIF data there. And there it was, along with creation and last edit date kept intact by Photoshop (when someone was adding the caption).
Neil was banging his tunes on 11th of April 2006, probably. I have been living a lie.
P.S.: I'm going to keep celebrating it yearly on 13th, though. Also, there's still a chance the tags are wrong anyway.
P.P.S.: Thanks to @upthorn for sparking my curiosity today.
Neil (actual, April 11th)
Neil (observed, April 13th)