The activation of the starter motor starts with the key (or push button in the early cars), but the most important function is carried out by the solenoid. The current used by the starter motor as it cranks on the engine is very high and the function of the solenoid is to act as a very heavy duty switch.
The key or pushbutton is very low current and only sends a voltage to the solenoid that turns on the real voltage to the starter. The solenoid is mounted in the engine compartment and typically either works or doesn't.
But we have often seen that the high current part of the solenoid begins to draw too much current internally and thus robs the needed current to the starter. The starter thus doesn't turn fast enough! Cooper S variants had the solenoid in a 12 o'clock position and the mini plus coopers is in the 12.10 position.
For original push button type solenoid, see BMK1727.