When did MMA records begin?

As long as a "record" for a fighter can be verified in respect of the fights he had, and providing it was under a "Mixed martial arts rules" format then it is recorded of which Sherdog is effectively the now MMA equivalent of Boxrec. As already said these can only be done retrospectively as MMA as a sport is still very new. Even with the first UFC's it was not referred to as Mixed Martial Arts and wasn't marketed as such.
The fighters that participated in these tourneys generally came from singular disciplines and their records within that reflected such. Hence why those fights don't count towards their MMA record. Case in point Bas Rutten.
It was only as the sport developed and relevant promotions became more organised such as Shooto Pancrase Rings etc that records started to be recorded in a common format that was accessible to everyone. If you have to put a date on it the mid to late 90's is probably the most accurate.