When referring to synchronous generators, what do they mean by "operating at no load or at full load"? Can I connect a load that is larger than full load?
Of course you can connect a load that's larger than full load.
The data sheet tells you what happens at full load. Unfortunately, the data sheet probably doesn't tell you what happens at more than full load. You would have have to do the experiment to see whether it failed to self-excite into it, exploded, blew a fuse, tried then shut down safely, or how long it took to overheat.
There is a school of hobby engineering that assesses maximum ratings by ignoring the data sheet and turning it up until it explodes, then backing off a bit. Sounds like you adhere to this teaching. Professional engineers tend to stick to the data sheet. Often the manufacturer has spotted something in the months or years of development and qualification that the user hasn't yet.
It is the maximum amount of VA flowing through the equipment at full load under specified conditions.
Rated load/VA is the load specified in the nameplate of the instrument as specified by the manufacturer. This is also the rated current that a machine can carry without any damage.
The maximum VA under nominal or rated voltage and power conditions. This is mostly the same as rated current.
Note: if you exceed Maximum load, depending on how much, a number of effects my occur.
The load is increased --> current draw is increased BUT there is still electromagnetic headroom. Generator heats up faster than literature states as it is now operating outside of rated condition
The load is increased --> current draw is increased but this time it is beyond the EM capability. EM stall starts to occur: reduction in frequency generated, reduction in main exciter if via a wound rotor (maybe with a main exciter for XFMR coupling).
With a synchronous generator, it is important to distinguish between full-load current and full-load power. Power = Voltage X Current X Power Factor. Neither the full-load current rating nor the full-load power rating may be exceeded without adverse consequences. That means that the power factor can not be lower than some minimum rating when the current is at the maximum rating. If a single-phase generator is rated 400 volts, and 10 kW at 0.8 Pf, the maximum current at 0.8 pf would be 10,000 W / (400 V x 0.8) = 31.25 A. If the power factor is 1.0, the maximum current would need to be reduced to 25 A to avoid exceeding the maximum power rating: 10,000 W / (400 V x 1.0) = 25 A.
The maximum current is the current that the generator can safely carry without overheating. The maximum power may be determined by the rating of whatever prime mover is driving the generator. The excitation and factors affecting the stability of the generator may also be limiting factors. The limiting factors can be found by examining the generator performance curves.