Which would be the "original" original, since he "originated" both songs?

Well, if an artist re-records their own song, that does complicate things. I was just trying to distinguish between what I called the "original" version, meaning that by the song's writer (but not necessarily - see below), and a "definitive" version, which might be by somebody else. I can't for the moment think of a proper real-life example, but perhaps Irving Berlin recorded "White Christmas" before Bing Crosby (I don't know whether he did or not, but let's assume he did for the purpose of this discussion) - then Berlin's version is the original, but in the eyes of many, Bing Crosby's is the definitive version, i.e. the one that people first think of when they hear the song title.

Now let's say Crosby recorded the song twice. One of those would be the definitive version (assuming you think it is anyway!) and the other would not be, because it's just another recording by the same artist (e.g. a live recording from a TV show or similar).

Sometimes a song is recorded by another artist before the writer records it him/herself. In that case it's possible for the "original" version to be by someone other than the writer, because it was recorded by the other artist first. Again I can't think of a real example for the present, but I'm sure there must be some. Well, maybe I can - I remember Ellie Greenwich wrote many hits in the 1960s with Jeff Barry, Phil Spector, etc., and many people would say that the original hit versions (by the likes of the Ronettes, the Crystals, Shangri-Las, Ike & Tina Turner, and many more) were both the 'definitive' and 'original' versions. Ellie recorded many of these herself on a 1970s (IIRC) album. Her own versions would probably not be called either the 'original' or 'definitive' versions by many people (but 'definitive' is definitely subjective; 'original' isn't if you accept my definition - I said 'if'!)

I have never heard it, but some friends told me that in the sixties, the Fuggs used to sing "Hey, Hey We're The Fuggs" and it was definitely better than the Monkies' version

Ah, that makes it a different song! ;-)

OK then, here's a festive one: "Wombling Merry Christmas" - definitive version is that by The Wombles, IMO :-)