Final Update...
I think what I have done will fix most examples of this problem, and will be quicker and much cheaper than a re-con starter, or getting your own starter repaired by a specialist.
I bought a new battery which did not make any difference, but it was cheap, at £50 inc from autobits .com in Farnham.
I removed the starter, from underneath, you need to remove the bottom engine cover, the top 1/2 of the dipstick guide (1 bolt then it pulls out) I also removed a couple of brake pipe brackets which you may or may not need to do), the elec. connectors to the starter (make sure you disconnect from the battery 1st). I used a 14mm ring spanner to remove the 2 bolts, and eventually managed to extract the starter. This is not easy as there is limited room.
Opening the cover of the solenoid reveals the plunger which just pulls out, and the 2 terminals, which were obviously worn. I cleaned the plunger contact up with abrasive paper, and ordered 2 new terminals (< £16 inc ) from Toyota (Lexus only sell the whole solenoid assy).
Tip - If you tell them the car is off the road they deliver to the garage next day (no-one seems to keep them in stock).
My starter mechanism had the Toyota part no on it (you need to quote this, but essentially its the same as on a Supra).
Replacing the contacts takes about 5 mins. and is easy - just make sure you put all the insulators back!
I also tested that the starter motor worked (I recomend you try this in a vice, or clamp it in a work bench). I connected it to the battery, using jump leads, earth to the motor case, +12v (to the teminal that connects the solenoid to the motor) and it whirred away nicely. Tip - connect to the starter 1st and then to the battery, this will produce less sparks. Mine was fine but its probably worth finding out before you put it all back in again.
I replaced the starter, easier than getting it out as the practise in finding the exact position you need to be in to get at the bolts when extracting will make a big difference.
I replaced the dipstick guide, the brake fluid pipe brackets, and engine cover, re-connected the battery and the car now starts perfectly, 1st time every time.
It may be possible to replace the solenoid terminals without removing the starter, but there is not much room, so it would be v difficult to see, there are a couple of very thin wires you could break under the solenoid cover, (1 is attached to one of the teminals) and you need to make sure that that you have put it back together so that nothing is shorted, so I wouldn't recommend it.
Tools I used were
10, 12 and 14mm ring spanners.
10, 12, and 14mm sockets, extension bar and ratchet.
plus a smaller spanner for the solenoid cover plate (I can't remember the size but would guess about 6 mm)
And a cross head screwdriver
I hope this helps someone some day.
If you do try this and get stuck, or this helps you then send me an e-mail!
If you try this and break something - don't blame me!