The most accurate answer you can get is from me, who named both.
Originally, we were mere players on Acclaim's version of BOTS. However, we were hackers who basically killed it off, then took it for ourselves (yeah, our morality has gone up a bit since then).
The main hack site at the time was gamecheetah, which refused to staff me even though my hacks became the only used hack in the game. I dominated the BOTS hack market (similar to megaarkade, except we aren't ***s like he is).
In response, we made our own hack site, and named it BotsCheetah, a reference to gamecheetah, but a hack site made exclusively for BOTS. Gamecheetah's site lost its BOTS users and inevitably (probably unrelated, I'm not saying we caused this) went out of business.
Auron released a WIP BOTS server on our hack site, which we took and created this game from. We then created an alternate website with a similar name, but used BoutCheetah as a partnership to BotsCheetah. At the same time, we hacked BOTS to the ground, and used it as an advertise pool, and eventually when it shutdown, we shutdown BotsCheetah, making BoutCheetah our main website.
A short history of our beginnings, and the end of BOTS.