Then I have moved coins around from different wallets and exchanges.
When you have coin on an exchange you dont really hold it but exchange does. So you dont get any forks that way. When you withdraw from exchange exchange will send you just the coin you are asking for and no forks of it.
If you hold coin in your wallet, then all forks are there. You dont need to claim anything. It is in your wallet. If bitcoin goes to zero in 10 years time and that fork is then new bitcoin you have it. If you play now then you might not have it anymore.