Are there any national regulations or internal airline rules limiting the number of go-arounds before the flight must be diverted to another airport ?
You tagged your question with airline-operations, which suggests you ask for air transport operators and pilots.
No maximum from regulation
Regulations impose no limit. It a matter of available fuel, crew professional judgment and operating procedures set by the operator.
Of course landing is limited by the aircraft certificate, based on flight tests by the manufacturer. If the cross-wind at an airport is above what was demonstrated during the tests, the crew will look for an alternate airport, or wait.
No maximum from standard operating procedures
Operating procedures may give general guidelines, but:
Setting a maximum number of attempts would encourage pilots to land on the last allowed one, which is counterproductive for the safety of the operations.
Why would landing again at the same airport be more dangerous than trying at another one with likely the same weather conditions and perhaps a shorter runway?
If there were a limit how would crews manage landings at isolated airports?
From Do multiple go-arounds on the same flight happen in commercial aviation?
None of four airlines I worked for (two commuters, two Part 121) had specific guidance as to the max number of go-arounds. However, as mentioned in the other answers, 2 or 3 was believed to be a good rule.
Flight is supervised from the ground by a dispatcher
Carrier flights are managed by a dual team: The crew in the cockpit has the responsibility for the aircraft and the dispatcher on the ground supports the crew and is the link between the crew and the commercial team. From What is the role of the flight dispatcher? How do the dispatcher and the flight crew interact:
Once the flight is airborne the Flight dispatcher keeps an eye on that flight and makes any recommendations to the Captain regarding any change in route, altitude, speed, destination or anything else relating to flight safety and the efficient operation of the flight. This is called "flight following".
Thus the crew is not alone, and if they appear to be tunneled into a get-there-itis syndrome, the dispatcher can talk with them about the best option for landing.
For an airline, the commercial division is the one under pressure
For a passenger transport operator, that's not the crew which has the largest interest in landing at the scheduled airport, this is the commercial division. A diversion means repatriation to the scheduled airport, missed connections, accommodation expenses, customer dissatisfaction, financial compensation, etc.
The article you cite, and the FAA unknown statistics likely refer to PPL pilots, unlikely to CPL and ATPL pilots. The article may also be excessive in their formulation with sentences like "For pilots, it's not uncommon to experience the desire to reach their destination, no matter what. This phenomenon is known as "get-there-itis," and it can be incredibly dangerous" and "Pilots who experience get-there-itis may disregard weather warnings, exceed aircraft limitations, or make risky decisions that put themselves and their passengers in danger".
