Hi,
1) As above... you can't disconnect the battery. That's a old test, and it really doesn't tell you anything.
2) Your battery is probably going bad. If you are going to keep the ski, buy a new one. A weak battery is hard on the starter, and hard on the charging system.
3) To test the system, use a volt meter, and just hook it to the top of the battery. Start the engine, and bring it up to 5000 RPM. You should have 13.5 to 14.5v. If you don't, then there is a good chance that the regulator/rectifier unit has gone bad. It is in the gray electrical box. It is square shaped, and has 4 wires coming out of it. On a seadoo, it is very rare that anything else goes wrong with the charging system. I guess I should add... Don't overheat your engine if you do this out of the water. You can only run the engine dry for about 15 seconds.
With that all said... please don't get confused. The original question of the "Alternator" is just semantics. By the purest terms... yes, you have an alternator. In the magneto housing of the engine, there are power coils, that produce alternating current. (on the yellow wires) From there, the "Rectifier" changes that to "Direct Current" (DC) voltage... and the "Regulator" keeps the voltage at a usable level. (around 13.5 volts) As I already said... your Regulator, and Rectifier are in a single unit. The word "Magneto" will get used with a charging system on watercraft, and motorcycles, because many of them don't have an external alternator. In the purest terms, a "Magneto" is an unregulated power source. The reason a SeaDoo has a "MAG" is because the ignition is a magneto/CDI system. (for most of the engines) The Mag coil puts out an unregulated voltage, and can reach 300v. The high voltage is needed to charge the capacitor in the CDI ignition.
Sorry that got long... but I wanted to give you the EXACT info, that may have been confusing you.
Anyway... check the output with a volt meter, and let us know what you get. But, it really sounds like you need a new battery. (that may be the only problem)