There’s no such thing as an American and a British “quite”. Both senses of the word can apply depending on the context and the intonation the speaker uses. Eg. It’s quite good. Vs It’s really quite good.
Anyway, I’d say that in Hebrew de leans more towards the “a little” side, while lemadai למדי is more on the “substantially so” side.
הוא די טוב He/It is good, but not great.
הוא טוב למדי He’s really pretty good.
Except this speaker is pronouncing it as the first word "die" instead of "quite" so I am still confused about how/when to use it. I hear people use it like "stop" it's enough already!! But there is no difference in how the speaker is saying it for being "quite tired" to my ears any way.
Tired, but not sleepy? I put “I am rather sleepy.” And that was not accepted, I had to put “I am rather tired.” To be fair, there is a subtle difference between the two. In English “tired” can mean physically worn out and needing to sit and rest, but not needing to sleep, and it can also mean being ready to go to sleep (sleepy).