Thanks for all replies and suggestions.
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Gord Baker - I don't have a fresh battery, but made sure it was fully loaded by hooking it up to a battery charger and checking that it was 12+ Volts after charging. Plus, when shorting the two big poles of the solenoid, it did start and I could use the Toro without a problem (until it needed starting, again).
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bentrim - Both. So, when turned to ON, I hear the solenoid in the carburetor, and when turned to START I hear a click in the solenoid between battery and starter. But no voltage on the starter pole of the solenoid. All safety brakes and stuff work fine. Thanks for the documentation.
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Shady oak - I do not have a booster charger. Battery tested fine with voltage meter, though. (It is not that old, either. About a year or so? Two at most).
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outdoorpowermike - I have not tested the battery with the motor running, but when shorting the solenoid, I can start it. Shouldn't that mean that the battery is good? It certainly means that the starter is good, I'm happy to say.
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mcspeed - I don't know if that is what you should take away from this thread, really. I have use this particular model for some 7 years on a big, rather rough terrain without a problem. Sure, I had to replace two of the spindles and of course the cutting blades (twice), but to me that is normal wear and tear. Nothing wrong with the engine and electrical system, up until now. And I suspect changing the solenoid will solve that problem. It takes a bit of time finding out what and how to trouble shoot, which is why I am very grateful for this particular forum and the videos that are out there on youtube.
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dpscott - The safety switch on the seat should not have any influence on getting the motor to run. It serves as a safety measure for the PTO only. (And it is still working properly).
Thanks again, folks! I'll keep you posted.