OK, the new starter went in yesterday and on the way home from the dealer I tried everything that would cause it to play up. Nothing, did not play up once.
This morning when I started the bike it fired right up way faster than before and it actually felt like I have a new battery (which I have). Even the cold starts are faster now. I did not realise how the starting performance had deteriated over time.
I'll see how it goes but so far it seems to be fixed.