Solenoids can be designed in a variety of ways and all accomplish the same basic task. Some use two small terminals and require a positive and a negative and both of those can be controlled in different ways. For example you could send a positive to the solenoid through a key switch and push button and supply the negative through a seat switch, mower deck switch and neutral switch such as used on lawn equipment. Some are grounded internally and need to be bolted down to operate. Some even only require a ground to one small terminal because the positive small terminal is supplied internally from the battery cable on the large post. When a negative is brought it to the small terminal the solenoid energizes.
Some systems even use a solenoid to bring in a larger solenoid on things like larger starter motors. Another thing to mention since this pertains to starters is you should never beat on your starter because it may be a permanent magnet design. When starter brushes wear or hang up in their holder they wont make good contact with the armature so people will sometimes bang on the starter and it will start working. You can sometimes get away with it but sometimes it breaks the magnets and they collapse in a crumbled mess onto the armature. The magnets are glued into the starter housing.