OK, just to make sure I understand you correctly: you turn the key to the start position, hold it in the start position and there is a delay before the starter begins to turn. Is that correct?
If so, I'd suspect either the ignition switch or the wiring from the ignition switch to the starter. The fact that you made several attempts with no action then it suddenly started would indicate to me that the ignition switch is more likely than the wire.
Here's how to test this.
The starter has two wires - the big fat wire from the battery and the smaller wire (think it's red/black).
The smaller wire should have around battery voltage (12v or so) on it anytime the ignition key in the Start position.
Pull the smaller wire off the starter. Take one lead of your multimeter and attach it to that wire. Connect the other lead to a good ground or battery negative terminal. Set your multimeter to Volts.
Push the clutch pedal down (or auto trans in park) and turn the ignition key to Start. Note the car will not start with that wire disconnected, but put it in neutral just to be safe! The clutch pedal must be down as the car is not supposed to start unless the pedal is depressed.
You should see around 12v on your multimeter. Rotate the key back to Off, then again to Start. You have to go all the way back to Off before it'll let you access the Start position.
You should see 12v every time the key is at Start and the clutch pedal is depressed.
If you do not ever get 12v or if it takes multiple tries to get the 12v reading, then you need to determine if it's the switch or the wiring between the switch and starter.
Pull off whatever steering wheel trim is needed to access the backside of the ignition switch. According to my 2003 wiring diagram, there should be a red/gray wire (manual trans or red/black for auto trans) attached to terminal 50 of the switch. Leaving the connector attached to the ignition switch, back probe that wire with one multimeter lead, connect the other to a good ground.
Repeat the clutch down/key to Start procedure.
If you now get 12v every time the key is in Start, then the ignition switch is doing what it's supposed to do. The fault then lies in the wiring between the switch and starter. This is a bit complex as there is a clutch switch and a relay involved for manuals, autos have a park/neutral relay. You'd need to check continuity on the various wires, verify the clutch or park/neutral switch was working, etc.
If you still don't get 12v or it takes multiple tries to get 12v, then you've got a failed ignition switch.