I think the perceived wisdom on HM's relationships with American presidents is that she got along very well with Reagan. It's hard to see HM heading out on horseback for a morning ride with any of the other US presidents she's met. Mrs Reagan has also always been very complimentary about the Queen and how well they all got on.
As Princess Elizabeth, President Truman welcomed HM to the US and, while standing next to her, told his audience that he was only disappointed that all Americans wouldn't get the chance to see her as he had been told that as soon as anyone became acquainted with her they "fell in love with her". It can be seen here:
I've got a soft spot for this hand-written letter from HM to President Eisenhower
in 1960 which included a recipe for drop scones that she had promised to send him while he was at Balmoral. It's so incredibly informal, telling him that she's tried the recipe with golden syrup and treacle, and that when making them for fewer people she just puts less flour and milk in. The letter is very sweet and suggests that they at least got on pretty well.
HM also seemed to enjoy the company of Mrs Obama on her visit to the UK. The Queen put her arm around her, which I don't remember seeing her do before or after.