A “Texas death match” is a wrestling term used to advertise extreme wrestling, first cited in 1955. Early advertisements stated “no fall limits” and “no time limits.”
 
 
Wikipedia: List of Professional Wrestling Types
Many types of matches, sometimes called “gimmick matches” in the jargon of the business, can be found in the performance art that is professional wrestling.
 
Some of them are especially crowd pleasing and occur relatively frequently, while others are developed so as to advance a storyline, and thus, such match types are used rarely. Specific match types in professional wrestling are often notable due to either their frequent use, logistics of setup, or a memorable instances of such a match. Often, specialty matches are used as a finale to a popular or infamous storyline.
(...)
Hardcore-based variations
Hardcore wrestling is a subset of professional wrestling where some or all of the traditional rules do not apply. Most often this simply means there are no disqualifications, which itself eliminates the countout, allowing decisions to take place anywhere. In fact, many so-called “hardcore match types” are merely euphemisms for the same sort of “rule-less” match, with the name meant to emphasize certain aspects of hardcore wrestling or to point out one specific difference. The most common euphemism is the Deathmatch, often with some kind of prefix (i.e, “Texas Deathmatch”, “Lights Out Deathmatch”) used to promote the supposed risk of injury wrestlers may sustain during the match.   
 
Urabn Dictionary
1. Texas Death Match
A Texas Death match is a wrestling match similar to the Last Man Standing Match, the only difference being that you must be pinned, then a 10 is inniciated. If the person pinned fails to get up by the count of 10, the participent loses the match.
At WCW Halloween Havoc 1993, Vader defeated Mick Foley in a Texas Death Match.
by The Kris New Jersey Apr 11, 2007
   
2. texas death match 
a gimmick match in pro wrestling
In a texas death match:
1) there is no disqualification;
2) pin/falls count anywhere in the building; and
3) a wrestler has to touch all four ring posts in order for a pin/fall or submission to count.
by Super Guy Feb 4, 2004
       
18 March 1955, Aiken (SC) Standard and Review, pg. 5, col. 6:
Graham’s last appearance in Augusta brought him a victory over Dizzy Davis in a Texas Sudden Death match.
 
20 February 1956, Fort Pierce (FL) News-Tribune, pg. 3, col. 2 ad:
WRESTLING
MAIN EVENT
TEXAS DEATH MATCH
TO A FINISH    
 
2 March 1959, Lincoln (NE) Evening Journal, pg. 13, col. 8:
Omaha—Japanese bad men Mitsu Arikawa and Tosch Togo whipped Dick The Bruiser and Ernie Dusek in the 7th fall of a Texas death match here.
 
20 January 1960, Winnipeg (Manitoba, Canada) Free Press, pg. 26, col. 1 ad:
WRESTLING
WINNIPEG ARENA
MAIN EVENT
TEXAS DEATH MATCH
NO TIME LIMIT
 
2 May 1961, Lincoln (NE) Evening Journal, pg. 23, col. 1:
They’ll meet in the main event of Promoter Jack Pesek’s mat show—a Texas Death Match, with no time limit and no fall limit.
 
16 January 1964, Holland (MI) Evening Sentinel, pg. 15, col. 3:
Under the rules of the Texas Death Match, submission holds only will count and the match will continue until one or the other admits defeat.
 
21 June 1970, Van Nuys (CA) Valley News, “Death Match Scheduled Tomorrow at Ventura,” pg. 17A, cols. 1-2:
In a Texas death match, falls do not count and the match ends only when one team can no longer continue.
 
30 September 1970, Port Arthur (TX) News, pg. 14, col. 3 ad:
MAIN EVENT
TEXAS DEATH MATCH
No Falls—No Time Limit
   
29 January 1972, Winnipeg (Manitoba, Canada) Free Press, pg. 55, col. 5 ad:
WORLD WIDE WRESTLING
MAIN EVENT—TEXAS DEATH MATCH
NO DISQUALIFICATION—No Time Limit—Falls Don’t Count
   
3 February 1983, Miami (FL) Herald, pg. 2F:
Ron Bass defeated Angelo Mosca in a Texas Death Match, the main event of a professional wrestling card before 2,284 fans at the Miami Beach Convention Center.
     
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From: .(JavaScript must be enabled to view this email address)
Date: 14 Sep 92 14:45:33 GMT
Local: Mon, Sep 14 1992 10:45 am
Subject: What’s a Texas Death Match? 
 
I forgot, what’s a Texas Death Match? 
     
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From: .(JavaScript must be enabled to view this email address) (Don Del Grande)
Date: 5 Feb 1995 21:42:30 GMT
Local: Sun, Feb 5 1995 5:42 pm
Subject: Texas vs. Mexican Death Match
 
I’ve heard the terms “Texas Death Match” and “Mexican Death Match” used.  Is there a difference? 
(Back in the 1970s, I always heard it called just “Death Match”, and it went like this: falls didn’t count, but after every fall, there was a 60-second break, and a wrestler won if his opponent couldn’t answer the bell (after a 10-count) to begin the next fall.)
 
Google Books
The Least You Need to Know
stories by Lee Martin
Sarabande Books
1996
Pg. 166:
“A Texas death match: no holds barred, no disqualifications.”
   
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From: .(JavaScript must be enabled to view this email address) (Scott A. Munro)
Date: 1998/04/01
Subject: Re: Texas Death Match? 
 
>On Wed, 01 Apr 1998 10:06:57 -0500, Geek

wrote:
>>What exactly are the rules for a texas death match?
>Falls do not count.  After a fall, losing man has a count of ten reach
>his feet, otherwise the match is over.
 
In the WWF, back in the early 80s, I believe they called no-DQ matches “Texas Death Matches.”
 
Google Books
Have a Nice Day:
A Tale of Blood and Sweatsocks
by Mick Foley
New York: Regan Books
1999
Pg. 356:
Actually, it was a Texas Death Match, but, being in Disney, a few concessions had to be made. In Texas Death Match rules, pinfalls count, but they don’t end the match. The wrestler has ten seconds to answer the bell. The match continues until one man can’t answer the bell.
 
Google Books
The Funk Family
by Ross Davies
New York: The Rosen Publishing Group
2001
Pg. 13:
Texas Death Matches. A Texas death match doesn’t end until one wrestler can no longer continue; after each pinfall or submission, the match goes on until a wrestler concedes defeat. Taking advantage of the stamina he had developed from his amateur wrestling days, Dory Sr. was nearly unbeatable in these matches.
 
Google Books
Wrestling’s Most Wanted
by Floyd Conner
Washington, DC: Brassey’s
2001
Pg. 11:
In a death match, the action continues until one contestant is incapacitated. For instance, the Texas death match ends when one of the wrestlers is unable to get up before a ten count.