December 1981, if you want the earliest defensible date. July 1983 if you want the date it became a thing you could buy. And the interesting part of the answer is everything in between, because that gap of nineteen months is where the whole genre was actually built.
Precision matters here more than it usually does, because thrash has an unusually short and unusually well documented founding period. Almost everything that made it possible happened inside about three years, and it can be walked through month by month.
The earliest defensible date

December 1981 and Welcome to Hell. The record is not thrash by any technical definition and it is where the clock starts, because it removed the assumption that a heavy band had to be able to play before it was allowed to make a record.
You could push earlier. March 1979 and Overkill is a serious proposal. But Motorhead's influence was ambient across all of heavy music, whereas Venom's was specific and detonated a particular set of bands in a particular two year window.
1982: the year everything moves
March. Rampage Radio goes on air at two in the morning on KUSF in San Francisco, the only outlet in America that will play an unsigned band's fourth generation demo tape.
June. Brian Slagel releases Metal Massacre on a label he has invented for the purpose, carrying Metallica's first released recording with the band's name misspelled on the first pressing.
July. Seven songs are recorded in southern California and start moving hand to hand through the post as No Life til Leather.
Later that year. Accept release Restless and Wild in Germany with a song on it that technically qualifies as thrash and is treated by its makers as a novelty. Venom release Black Metal and name a genre that will not exist for nine years. Anvil release Metal on Metal in Canada and Lars Ulrich and Scott Ian take it apart bar by bar.
By the end of 1982, every component exists and is circulating. Nothing has been assembled and published.
1983: the year it becomes a genre

February. Metallica leave Los Angeles for El Cerrito, four hundred miles north, because a room full of people in the East Bay has reacted to them in a way nobody in Los Angeles ever did.
April. Dave Mustaine is dismissed in New York and put on a bus home, and forms Megadeth within weeks. Kirk Hammett is recruited out of Exodus, and Exodus hire somebody else and carry on.
July. Megaforce releases Kill 'Em All, financed by a second mortgage.
December. Metal Blade releases Slayer's Show No Mercy, paid for out of a respiratory therapist's wages and a loan from a guitarist's father.
| Date | Event |
|---|---|
| March 1979 | Motorhead, Overkill: the tempo and the double kick |
| December 1981 | Venom, Welcome to Hell: the permission |
| March 1982 | Rampage Radio begins on KUSF, San Francisco |
| June 1982 | Metal Massacre, first pressing, name misspelled |
| July 1982 | No Life til Leather demo recorded |
| 1982 | Accept, "Fast as a Shark"; Venom, Black Metal; Anvil, Metal on Metal |
| April 1983 | Mustaine dismissed; Megadeth formed; Hammett leaves Exodus |
| July 1983 | Kill 'Em All released |
| December 1983 | Show No Mercy released |
| 1986 | The form reaches its finished state |
So what is the date
If somebody wants one date and will not accept a range, use 25 July 1983. That is the day the complete form went on sale.
But the honest answer is that genres do not start on days. What happened here is that a set of conditions came together over about three years, and then a music that had been latent inside those conditions appeared in four places nearly at once. The date on the sleeve records the moment the thing became purchasable, which is not the same as the moment it became real.
The word itself, incidentally, arrived later than any of this, and the story of where it came from is in Why Is It Called Thrash Metal?. The full account of the period is in where this sits in the story.
Why the founding period is so short
Three years from the first stirrings to a finished genre is fast even by the standards of popular music, and the compression has a specific set of causes.
The first is that all the parts already existed. Nobody had to invent a technique from nothing; the tempo, the riff vocabulary, the beat and the attitude were sitting in four separate places waiting to be combined, and combination is much quicker than invention.
The second is the postal network. A demo tape could cross a continent in a fortnight, which meant a band in New Jersey could hear what a band in California had done before either had a record out. Every scene in music before this one moved at the speed of touring; this one moved at the speed of the mail.
The third is that the people involved were between sixteen and twenty two and had nothing else on. A genre made by professionals with careers takes a decade. A genre made by teenagers in garages, all of whom are trying to outdo the tape they got last week, takes about thirty months.
When it finished
The other end of the question is easier and less often asked. Thrash reached its finished form in 1986, and after that everything is elaboration.
Between March and October of that year the genre released the most ambitious record of its American wing, the most extreme record anybody had made, the heaviest record of its German wing and the sharpest politically. Every technical question that had been open in 1983 was answered in those eight months. How fast, how long, how dry, how ambitious: all settled.
That is why the founding period reads as three years rather than as a decade. The thing was started in December 1981, assembled by July 1983, and completed by October 1986, and everything since has been people working inside a form that was finished before most of its current audience was born.
Sources and notes
Dates refer to original releases and to the bands' published accounts. The Mustaine dismissal is dated 11 April 1983. The first pressing of Metal Massacre is June 1982.