I'd say there are two scenarios in my world.
The record rides were either in organised time trials or as standalone record attempts.
In the UK, time trials are organised by CTT (Cycling Time Trials),
formerly RTTC (Road Time Trials Council). For technical reasons,
any records and championships still have the RTTC banner attached.
If I turn up at a suitable time trial, hoping to break a record then I'd
call that a record attempt.
It's always possible to turn up and unexpectedly break a record - but
I've never had that happen.*
The standalone record attempts don't have an organising body. Instead, the Road Records Association exists as a verifying and certifying body. The RRA provides a framework within which record attempts are made, and also maintains a list of recognised records. Related to the RRA (which can be thought of as a 'national' body) are a number of regional organisations, such as the Southern RRA, Western RRA etc.
It's worth saying, therefore, that for a standalone record attempt you need to do a bit of organising and arranging yourself. Although every ride has its own characteristics, you will need to find Timekeepers, Observers and Witnesses. The role of the Timekeeper is obvious. The Observer would typically try to keep an eye on all the proceedings and produce a ride report. Normally, the Observer would travel in a car following the ride and hence would be well placed to check compliance with all the rules. Witnesses are needed at any turning points and at starts and finishes. There's clearly a bit more to it than that, and the RRA can usually help with finding people to assist etc.
* It nearly happened for me & Dave in 1998. But only nearly.