The question that hasn't been asked, even though you were driving on relatively flat ground, was the motor being lugged at a low RPM in a tall gear? This will cause over heating in a diesel engine because the high torque will keep it pulling but that does generate a lot of heat. The exhaust gas temperature gauge that NP mentioned will tell you real quick if your driving technique is causing the problem, it will go up well ahead of the coolant temp because there is a lot of mass to absorb the heat within the engine block and heads and a lot of water to heat up as well.
As has been mentioned, the temp gauge and the light/buzzer are on different circuits and use different sensors, so it the gauge and the buzzer says it's hot, it is hot

Generally the light buzzer will not come on until the temp gauge has virtually maxed out, the difference between hot and real hot and you need to do something now type of thing ....
If you get the radiator core cleaned, be sure to fit a Tefba
https://www.ebay.com.au/sch/i.html?_nkw ... gKyafD_BwE or similar filter in the top radiator hose to catch any junk trying to get back into the radiator to block it up.
As far as that radiator cap, if you can find a cap with the correct depth between the cap clamping mechanism and the rubber to ensure the rubber does actually seat before the clamping takes effect ... that need to push down on the cap to get the lugs past the safety stop is a good indication. Next, see in you can find a cap with the rubber seal just under the metal cap that seals the top of the neck so water can only leave/enter via the overflow hose, then add a 2 ltr bottle at the end of the overflow hose with the hose a few inches from the bottom of the bottle to allow any garbage to settle and not be drawn back into the radiator.
If you don't have any aluminium in the cooling system, add some Phosphoric Acid to the fresh radiator water with no rust inhibitor type coolant added yet. Drive the rig around and watch the water temp, if it starts to rise and takes quite a while to settle back down, let the whole system cool down, remove the rad cap to release the pressure, then refit and remove the water filter plunger and screen to make sure it isn't blocked and the reduced water flow is causing the heating issue. Keep this routine up until the rust flakes stop appearing in the filter and the water looks clear yet the overflow bottle will have a heap of white sludge in the bottom.
Empty the over flow bottle regularly until the white sludge stops building up, check the filter to make sure there is no new rust flakes coming through, they might be white or have white edges, but they are still rust flakes. Be sure to run the heater water through the heater core as well (no need to turn the fan on unless it is actually cold and you want the heat) because this it a prime spot for rubbish to collect and stuff up a nice clean cooling system the first time you use it ...
Once you are sure everything is clean, this could take mths, drain the coolant and refill with the required antifreeze and rust inhibitor mix that is used to maintain a cooling system in good condition. Test the coolant with those PH strips every so often to ensure the coolant remains alkaline and does not drift into the acid realm, top up the inhibitor concentrate level as required to maintain the alkaline level that will prevent new rust forming and all will be well.
T1 Terry
T1 Terry