The part at the link you provided appears "visually" to be the same as my actual original part. However, the identification numbers seem to have no reference to applying to the Engine numbers I included in my post. The confusion seems to be that apparently ALL parts diagrams do not list the Solenoid as a separate item (separate part number) and is listed as part of the Carburetor units.
However, yesterday after my last post I found two different part numbers for a Solenoid that "hopefully" is a replacement unit for my engine. Those numbers are: KH-24-757-45-S and KH-12-757-33 with the last number being visually more like the original than the first. I was going to call Kohler to try to check and verify if these numbers ARE or are NOT the proper part for my engine. Today is a holiday so the call was not productive.
I think I will find an "after market" source and (since the price is nominal) go ahead and order a solenoid for the last number above. The OEM prices for these are rather expensive to the point of being almost like "legal robbery"
Thanks again for your help -- Vern....
Note: I like your Play comments. I am age 86 and still enjoy playing and keeping active.
Forgot to mention:
It appears the Solenoid malfunction is intermittent and not consistent. While the engine was running (at a normal idle) I disconnected the power connection wire and the engine stopped almost immediately -- so the coil definitely had the plunger" pulled open allowing the flow of fuel and the wire disconnect allowed the "plunger" to return in the closed upward position and shut off the fuel causing the engine to stop.