There is still some mystery here georgex. If the primer squirted the full amount of petrol for all three squirts of the bulb before you tried to start it, it should have fired for several seconds, unless the primer was squirting a lot of water in that petrol - so much that the water separated during the priming process, and shorted out the spark plug. To me it seems very difficult to mistake fuel that is as contaminated as this would require, for petrol when you poured it into the tank.
You now need to run it for quite a while on clean fuel before you store the mower. I bought a line trimmer a few days ago (a 30 year old Kioritz Echo straight shaft). The previous owner had found it wouldn't start after a couple of years laying in his shed, so he pulled the starter continuously for ten minutes until it did start. Then he ran if for a while and stopped it. It wouldn't start again after that so he left it in the shed for another six months then sold it - to me. By then the water from the tank that had found its way through the carburetor had rusted and stuck the rings. There was plenty of water in the fuel filter, which was why it was so hard to start (it forms a film across the filter), and water in the carburetor of course. He obtained a diagnosis from an expert, who told him it needed a carburetor kit and was otherwise OK. I think that expert must have looked at it from quite a distance, because the engine would not rotate by pulling the starter.
So, at this point your engine probably has water in all the wrong places. You will need to run it until it evaporates all the odd drops of water from obscure places like the valve chest and breather, then change the oil, or you might find a nasty surprise next time you try to use it.
