Just musing and not for Lizzette to worry about.
If the start battery was flat/shot AND it IS an energise to run solenoid, then judging by what we learned about the Shanks the stop solenoid may be a complicated but of kit with two windings and contacts. If so, the clicking can be easily explained by the flat battery, while the smoking (assuming it was the stop solenoid) could be explained by the flat battery not providing enough energy to fully move the solenoid and open the pull in coil contacts. If so, the low resistance, high currant pull in coll may well smoke if left energised for more than a few seconds.
Note - conjecture, not fact, happy to discuss.