The ignition switch switches a LIVE current down to the starter solenoid, the small terminal, they will nearly always work without load, but not on load when duff.
The solenoid could be sticking, esp if it is a BOSCH starter ?, connect the leads as said, then take a wire from the main battery connection to the small terminalk that comes from your ignition switch, and switch it manually, NB make sure car is in neutral, before trying!!, leave the ignition key on so lights are lit up(position 2) the engine should start, if it is a solenoid problem you will not hear a LOUD distinctive CLUNK as the ring gear isa engaged by the drive pinion on the starter.
Have found a few here the solenoid connector has been bodged, these need to be 100% perfect, ie a crimped on terminal will not cut it, needs soldering and heat shrinking, recently done a escort diesel with this very problem, new starter new battery new alternator, and it was the solenoid connection, snipped old one off soldered new one on(with adhesive lined heatshrink) and 100% perfect in the cold winter snow we had, even after leaving 2 weeks.
The problem particularly with bosch is the solenoid internals are coated with varnish, fine when brand new but a few years old and the varnish comes off and jams up the solenoid travel, so it can't travel full length, the first part of travel is to engage the pinion(gear) then the next bit is to actually bridge the terminals(inside solenoid) to apply power to the starter so it turns the engaged engine.
tabetha