I forget the percentage but what comes from your nuts is a small percentage. The majority is from the prostate.
Correct. Most of the semen volume comes from several glands inside your body, the greatest amount being supplied by the prostate, and only a small percentage from your testicles. Having big balls does not in itself mean a man will shoot a big load.
Volume appears to be a function of genetics, as well as age, and degree & duration of arousal. Young men produce more semen than older ones. And the sexual glands are analogous to the salivary glands in your mouth, so that like them, anticipation can get them flowing prior to orgasm.
Therefore, the longer and more intense the period of arousal, as with foreplay, the more fluid they will be producing. And if this fluid is not quickly released it will continue to build up in those glands, resulting in a greater volume when orgasm finally occurs.
This is the cause of so-called "blue balls" that younger men especially experience. It's produced by the painful pressure of excess unreleased fluid during prolonged sexual arousal. The pain radiates throughout the pelvic area, often giving an aching sensation in the testicles, when the source of the pressure may actually be elsewhere. Some of this fluid pressure may be felt in the epididymis, which is the duct directly behind the testes. Pain may also be experienced in the vas deferens.
The prostate is a small gland, however, and the other glands even smaller, and would themselves barely fill a wine glass. It's difficult to imagine how they would produce enough semen to fill one. I have to wonder if what the OP saw was the result of editing or other effects.