Generator wattage via their breakers
First no mater what the generator’s alternator has stamped on it this means nothing! You do not know the test conditions in either case for the alternator or the complete gen set. And taking things apart probably voided warranty.

I maintain very large gen sets and to provide an example one is a 350kva this gen set is certified for life safety equipment and I have loaded in one extreme case close to 500k it was not a life safety issue but we lost a secondary and we needed that amount of power 490a for close to an hour and the owner said to try it. It did fine. 

I have small 5-8k gen sets that can maintain 25% overload conditions but these were designed for heavy duty use 

My biggest home generator is similar to yours and I would not expect it to be able to pull more than full surge load any longer than ~15 seconds (the amount of time for most large fans to come to speed) why do I say this because the windings and slip rings are 1/2 the size of my smaller heavy duty models. Without knowing the testing that gen set was certified to it would be best to keep your loads to 80% of the values unless they are stated for use at 100%, my gen sets only the life safety ones are certified for 100% full load (3 + hour run time) with a 15 minute cool down after loading required.

Added per op comment:
The breakers are inverse time and will allow more amperage to be drawn than there listing based on the curves for that brand. 50*240= 12000w  the breaker will hold for a short time well beyond this value normally allow 3-5x for a short time.

code requires the wiring to be sized 125% for motor loads and continuous loads most home loads of any size are not continuous but let’s do a standard de rate of a 50 amp device .8 x 50 = 9000w  that is close but all the loads that are non motor or don’t run for 3 hours or more are rated at 100% so real value is someplace will be in between 9-12kw  

Code allows / requires the mfg values instructions to be used per NEC 110.3 in the U.S.  as I have explained some alternators may have some value that may have a different capacity than what is stamped on it. My gen set also has additional 120 outlets to fully load my gen set the fine print shows these receptacles must be used on loads that are isolated from the main, are they cheating ? The total it can output meets specifications.
 Most of the small generators play this game.

This game is just like shop vacuums and air compressors. If you have either of these things that are not true commercial units don’t look two far or you will get really mad because they don’t even come close to the HP rating they advertise. 
The only way to know on your gen set is to use a load that is within the testing requirements of the generator and measure the output.
I did this and following there requirements mine passed on a real 100kw testing system at my plant. But I had to have 12 amps on the 2 separate 120v receptacle sets. 
You can ask as I did for testing conditions (they even state sea level and 72 deg f temp and premium fuel  (no methanol) the methanol in the fuel I was using was the difference well that and having to pull 12 amps on ea leg through the receptacles. So it may sound funny but it’s not what you see it’s how things are measured.