You need to include your information about the seadoo you own in your avatar. Yes the battery needs to have a full charge to crank over the engine, and yes it needs the lanyard to crank over, other wise the starter won't get an power to crank the engine.
Here is some information on doing a compression test properly;
You'll need a compression tester. Go to Auto Zone. I think they sell for like $25.00. The compression gage, will screw in the cylinder head in place of the spark plugs. To test compression, remove both spark plugs. Place spark plug caps on the plug cap studs near the cylinder head to ground the empty caps. This completes the circuit of the ignition electrical system and prevents any electrical problems from the caps being un grounded. Using the correct adapter for the threaded end of the tester,( same length of the spark plug threads length)screw in the tester in one plug hole. Hold the throttle wide open. Push the start button. Watch the compression gauge, when it peaks out at the most compression, let go the start button. Read the psi number. I would do it 3 times to be sure it is accurate. Check both cylinders the same way. The ideal compression is 150 psi per cylinder. If it is less, it's not a problem as long as they are close to being the same. If the psi is less than 90 it might need be time for a tear down and a rebuild. If the psi in 1 cylinder is say 140 psi and the other is 80 psi you need to tear down and repair. This difference is a lot then there is a problem. I hope this helps you.
Karl