Dragon Magazine #63 established the AD&D 1e UA Barbarian
Your question touches on an assumption in terms of "what was an official source" for OD&D or AD&D material? The 1982 article by Gary Gygax (Dragon #63, pages 8-11) laid out the Barbarian (a sub class of Fighter) more or less as we ended up finding it in the Unearthed Arcana book for AD&D 1e. The linkage is explicit.
With that considered, White Dwarf was a publication from which numerous monsters were introduced that ended up in published books like the Fiend Folio, so you can argue that it was sufficiently "official" that it introduced the Barbarian to the game in a playable form, which means that ...
White Dwarf was probably "official" enough to be first for D&D
That would make @John Dallman's answer correct in terms of who got 'first' if that matters. But the Barbarian therein is different from than the one in Dragon #63 and UA. It is argued on the ad&d redit that the Original Barbarian in White Dwarf #4 was written for Original D&D, not AD&D, and I agree based on the evidence and presentation.
In 1977 people are still writing content & classes for OD&D. This
would be one of them. The AD&D Player's Handbook wouldn't even be
released for another year. Having a d6+1 HD progression would either A) give them a slight edge on the Fighting-Man (if using LBB 1d6 HD
only) or B) Slightly better than Cleric, but still less than Fighting-Man (if using Greyhawk HD).
It's likely "B", and done to preserve the importance of Fighting-Men
being a central class, seeing how Barbarian gains a number of extra
abilities like the thief. Making their HD also superior would be
pretty unbalancing along with double Dex AC and Con bonus.
FWIW, the RPG.net index also associates the Barbarian from White Dwarf #4 and #12 with Original D&D, not AD&D. (index.rpg.net entry on White Dwarf #12).
It began as a sub class, not a class, in TSR material
In AD&D 1e it was a sub class of Fighter in the TSR Unearthed Arcana version. In AD&D 2e it was a kit for the Warrior Class (Complete Fighter's Handbook, 1994) as were the Amazon, Samurai, Cavalier, etc. In Asbury's original White Dwarf article, Barbarian was explicitly introduced as a separate class, not a sub class of anything, which makes the White Dwarf case considerably stronger in terms of introducing the Barbarian as a Class.