"tout à fait" always means "completely", "absolutely", "totally" C'est tout à fait exact = It's absolutely true. Je ne suis pas tout à fait prêt = I am not entirely ready. C'est une bonne idée, n'est-ce pas ? - Tout à fait ! = It's a good idea, isn't it? - Absolutely! To my understanding, "quite" in English can mean "completely", but also "to some extent". In French, "tout à fait" doesn't include the latter.

I am British, but I understand this as 'I am completely in accordance'... with whatever is being said. This also makes sense when I am speaking about myself, i.e., I am in accordance with myself.

Well, there are several "exact translations" into English: - I quite agree with you. - you're quite right. - Yeah, you're right. - You're absolutely right. - I couldn't agree more. - You nailed it. - I agree. - I quite agree. - I completely agree. - I absolutely agree (with you) (on that). Any of these (and no doubt others) would fit the bill: it just depends on the context, the level of formality, and who you are speaking to... The French expression that you give is more flexible and and can be used in a wider range of contexts.