A charged battery, after coming off the charger and sitting for 10 minutes or so will read 12.6v. Realize that most multi-meters will read a couple tenths of a volt off...either high or low. If you connect your multi-meter to a known good battery with a full charge, the battery WILL be 12.6V. That's the physics of the lead acid battery. So...get an idea of how your multimeter reads from reading a good charged battery. Remember the difference, as it is important when checking the control cutout. .1v could be significant. For example...if your multimeter reads 13v on the charged battery, you must remember that your meter reads .4v high, and subtract .4v from all your readings.
Back to your regulator. With the cutout closed, your generator should produce 13.5 to 14.4v. This is controlled by the voltage regulating contact in your control box. 13.5v is what you would like if you drive the car frequently and rather long distances, as it will not overcharge the battery and boil out your water. 14.4v charge would be for those who rarely drive their cars often or far. The higher voltage will charge the battery faster, but on a long trip will boil the water out rapidly.
The cutout is a separate function. when the generator makes a certain voltage above the resting 12.6V battery voltage, it closes the contact so the generator is connected to the battery. That should be about 13V, give or take a point or two. Just as the voltage regulating contact can be adjusted for the charging voltage you want...the cutout contact is also adjustable.
We are finally up to your initial question..."should the cutout open during driving". As long as your generator and regulator are working properly, the answer is no. But, if your generator is not producing the 13v to hold the cutout relay closed, then it could very well trip open. Examples of this in normal driving would be:
1) At a low engine idle and high draw with headlights, fan and everything running, the generator cannot keep up with demand, so voltage falls to battery voltage and the cutout opens, not knowing any better. This is normal and not a worry.
2) You ask for more current than the generator can supply...say electric fan, headlights, high watt stereo system, phone charger, etc...then the generator falls behind, the voltage drops to battery voltage, and the cutout...still not knowing any better...trips open. In this case the system is behaving normally...but you are overloading the generator and could overheat it to death over time.
If you suspect a problem, I would pop the top on your control box and observe the 2 contacts while watching the accompanying battery voltage. Remember to adjust for your multimeter error. Before starting the engine, the voltage contact will be closed, and the cutout contact open. After start on a warm engine, the cutout will likely remain open if your idle is under 1000rpm. . Slowly rev the engine and note when the cutout contact closes...it must be more than 13v, but less than 13.3v. If it is not, adjust to bring it into this range. Next, once the contact closes, and with no extra loads on the car (headlights, radio etc) continue to slowly rev the motor until the voltage stabilizes. There will be a slow fluctuation as the contact cycles, but this must be in the range of 13.5 to 14.4v, as we talked about above.
If the test above is good, but you are still getting the red light and low voltage, then either the generator is loosing its "oomph", or you may be over-powering it above its rated load. Of course, if the test is bad, then the control box either needs adjusting or replacement. Best to double check all your connections and run a re-check, though, before condemning anything.