The primer on these motors does NOT squirt fuel into the motor. There is no choke on these motors.
It is only a (normally open) solenoid which opens a (normally closed) valve inside the primer assembly, allowing the fuel pump to squirt fuel into the intake, but only as long as the primer solenoid is activated (this means only so long as the key is being held in, which is holding the valve open, and the engine is being cranked). As soon as you quit pushing in on the key, the valve closes and shuts off fuel flow through the solenoid. Pumping it does nothing except cause the solenoid to open and close each time you push in on the key. You can hear the solenoid click every time you push in on the key. If you pump it without cranking the motor, you are doing nothing except making a clicking noise, as no fuel is going anywhere.
What you want to do when the engine is cold, is to push in on the key WHILE CRANKING the motor. The motor should start quickly if the throttle is in the fast idle position. You might need to pump the solenoid momentarily while cranking, until it starts to fire, but don't hold it open more than a couple seconds at a time while cranking, to avoid flooding.. As soon as it starts, let up on the key (quit pushing it in) so as not to flood the motor. If it tries to die, push in on the key for only a half second or so. You may need to do that several times until the motor starts to run on its own. You may also need to adjust the speed of the motor with the fast idle lever while it's trying to start and during warm up. Do not use the primer when the engine is warm so you don't flood it.
Normally the red lever on the solenoid should be in the run position which is as far to the right (clockwise) as it will turn. If your battery is dead and you have to rope start it, turn the red lever 90 degrees to the left (so it's at a right angle to the run position). That manually opens the primer valve. You will then need to turn it back to the run position as soon as it starts. You can also use the manual lever to start your motor in case the primer solenoid or its switch fails.
If your primer has a Schrader valve on it, the purpose is to hook up a can of "engine tuner" to clean carbon out of the motor while running it.
Be careful not to over tighten the screw on the strap holding the primer solenoid in place. That may cause it to crush the valve inside the solenoid. It should only be barely snug, not too tight.