I've got a Yamaha 9.9 tiller and the only way to kill it is to pull and replace the emergency lanyard. I seem to remember outboard motors that you could press in on the emergency kill switch and kill the motor.

Am I just making this up?

You are not imagining it. The outboards I recall from the 60s and 70s either had a button on the lower cowl or in the end of the twist grip throttle. The dead man lanyards replaced those.

I have an '05 Merc 15hp and it has the kill button on the end of the throttle

We just got a new 30hp Suzuki tiller outboard and pulling the lanyard was the only way I could stop the motor the first time we used it short of unhooking the fuel line. After reading the manual, I found the lanyard emergency kill switch is also the stop button. You just press it with the lanyard connected and the motor shuts off.

Are you sure you cant just push it? Not 100% sure about the 15hp 2 stroke Evinrude I sold long ago but on my old Tohatsu you push the red button to shut it down normally. Pulling the lanyard does the same thing but that’s for safety in case the operator falls overboard. You can still push the button with it attached.

P210 for the win. On a yamaha small outboard you just push in the red button to kill the engine. You do not have to pull the lanyard. If that doesn't work its a defect on the switch. FWIW one of the things Yamaha does well is owners manuals are easily available online if you don't have one.

I think you just push the red button. I have an older 15 hp Yamaha, looks very similar to current motors.

Even though the red button doesn’t move a lot(in) it kills the engine. I never really pull the cord.

I have a 15hp Mercury and there is a kill button at the tip of the tiller handle. I can kill the engine using that or the lanyard switch.