The Z Fighters are a sub-team of fighters that are part of the Dragon Team (ZZhan Shi, Zetto Senshi, "Z Warriors"). They have no formal leader or posts because they are simply a collection of martial artists that meet together from time to time. Their main purpose is to fight against other teams in order to prove that they are the best.
Here are some examples of events where Z Fighters were involved: The First Battle of Changan between Liu Bei and Cao Cao where they were led by Zhang Fei and Zongheng; The Second Battle of Changan between Liu Bei and Cao Cao where they were again led by Zhang Fei and this time also including Zhuge Liang; And finally The Third Battle of Changan where they were led by Zhang Fei yet again. In all three cases they lost but that doesn't mean that they weren't worth mentioning.
Now back to your question, it's not clear if they were called Z Fighters at the time of their creation or later so we can only guess that they were named after the team leader Zhang Fei. But since they were never given any official post or title afterwards, we can assume that they kept their name as it was before he created them.
Mr. Satan refers to the Z Fighters (ZZhan Shi, Zetto Senshi; sometimes known as the Z Fighters, Earth's Special Forces, or even the Gold Fighters) as a tiny squad of warriors on Earth. They have become the unofficial defenders of Earth due to the numerous conflicts they have waged in order to prove their talents and attain other aims. Mr. Satan also calls them his "bastards".
They first appeared in an episode of Astro Boy titled "The Brave Soldiers from Space!". The characters were designed by Yoshikazu Yasuhiko and originally voiced by Kappei Yamaguchi and Nobuo Tobita.
In the manga series written by Osamu Tezuka, the characters appear in almost every story arc involving space heroes. In addition, some episodes are set in the past when the characters meet up with Astro Boy or before they join him. The characters also have small cameo appearances in many other stories throughout the years. In total, there are over 100 episodes featuring the Z Fighters.
In more recent years, the characters have been referenced in various anime including Cardfight!! Vanguard, Gintama, Food Wars! Shokugeki no Soma, and Dragon Ball Super.
There is now a new generation of space heroes called "Space Heroes 2020" who also fight evil robots named "Borg" on behalf of humanity.
When the two Saiyans arrive on Earth, the Z Fighters are referred to as an actual organization for the first time; while the six warriors Krillin, Yamcha, Tien Shinhan, Chiaotzu, Piccolo, and Gohan are fighting the Saiyans, Yajirobe falsely claims credit for organizing and leading the group in a shameless attempt...
Z Fighter is also the name of a video game based on the anime series. It was released in Japan by Capcom on February 9, 1992 for the Super Nintendo Entertainment System. The game features eight characters from the series: Goku, Raditz, Vegeta, Nappa, Piccolo, Mr. Satan, and Moro with each character having three levels that can be played either alone or with a friend via link-up support.
In addition to these characters, four other fighters can be selected by the player at any time during the game: Eugei "Eugene" Bogard from Street Fighter II: Champion Edition, Rufus "Red" Rockerbell from Rock 'n' Roll Wrestling, Terry "Tiger" Bogard from Street Fighter II Turbo: Hyper Fighting, and Yun "Yang" Ji-Haeng from World Warrior Arena. These special characters can only be used once per stage and cannot be chosen again until after the current selection has been used.
The game was later ported to the Sega Genesis by Data East under the title Z Fighter. It was released in North America in August 1993 and in Europe the following month.