Verification code
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Anyway, I got lots of helpful advice on how to implement my two-step verification, with considerable detail from our honourable and learned leader (called A. Still). I tried this out and more or less got it right. Basically, you put in your email address and your usual password and a 'verification code' will be sent to one of your OTHER 'trusted' devices. This is a 6 digit number which you key in below your password.
I then noticed something that said you could dispense with 'two step-verification'. Guess what - I opted for this but got into another muddle in my Account Details via System Preferences>Cloud. So I rang my good friend Frank who lives in Kentucky and dispenses advice to Apple muddlers. He took me through the process bit by patient bit and, hey presto, I am now free of two-step verification.
If you want two-step verification, you can opt into it. There's quite a bit of advice on Apple websites including this one.