The requirement is for, "a flight of 300nm during which full stop landings at two airfields different from the aerodrome of departure shall be made". Consequently, any flight that covers at least 300nm during which two landings are made qualifies irrespective of how far apart the airfields are. As an extreme example: Lee-on-Solent - MID - GWC - SFD - MID - OCK - BIG - DVR - SFD - Bembridge - Sandown - Lee-on-Solent is about 305nm and will meet the requirement.
As for proving that the distance has been covered, logbook evidence is normally sufficient although, if you are claiming something odd, a copy of the PLOG might be useful.