Deputy Secretary versus Under Secretary?

Long time since I’ve been here. Alright, so I’m not familiar with it, so I thought I should ask. I know that in the UK they use under secretary quite often. In the US, it’s more of deputy secretary. However, I also know that there are some positions in the US government titled “under secretary.”

So, a few questions.

1) Between US and UK positions, what is the difference between deputy secretary and under secretary?

2) Between US and UK terminology, what is the difference between the two under secretaries?

3) Between the two US positions, what is the difference between deputy secretary and under secretary?

I realize that some of these questions come into conflict with one another, but that’s what this question is for. So, my fellow Fluthers, do what you do best. Many thanks in advance!