Also is this a list according to you or some rankings, survey etc?
Can't talk about the ten most reputable US universities and not include Stanford, Chicago and Berkeley. Dartmouth and Brown are great but indeed borrow very heavily from their Ivy League affiliation which tends to portray them in a far exaggerated standing than they truly deserve though Tuck Business School and both medical schools are highly reputable.
I would say the top ten in the US considering both undergrad, postgrad, national and global and peer/employer review rankings are (in no order):
1) Harvard
2) MIT
3) Stanford
4) Yale
5) Princeton
6)Caltech
7) Columbia
8) Pennsylvania
9) Chicago
10) Berkeley
But tbh it should really be a top 15 to include Dartmouth, Brown, UCLA, Duke, Michigan or 17 for Northwestern and Georgetown.
In the UK you are pretty much accurate though I would inclined to remove Durham from the list in favour Manchester. The former is great for undergrad but the latter has a far stronger global reputation even here in the US and in Asia. People on TSR seem to really been keen on Durham when in fairness except for a handful of undergraduate courses, it's really nothing special and if anything gets the attention because it's an old-ish university with a collegiate system like Oxbridge and hence is paraded around as some "Oxbridge lite" college.
Best 10 in the UK all rankings and reviews considered (in no order):
1) Cambridge
2) Oxford
3) Imperial
4) LSE
5) UCL
6) King's College London
7) Edinburgh
8) Bristol
9) Manchester
10) Warwick
Would be tempted to include Glasgow, Nottingham, Birmingham. The rest of the Russell Groups are really quite insignificant in a global standing. St Andrew's is great for undergrad and certain humanities subjects and of course borrows heavily from the fact a certain Prince went there.