AmE uses one-on-one in both a confrontational sense, or a face-to-face sense, so one-on-one meetings, lessons, calls, etc. occur widely in AmE.
I checked the Corpus of Contemporary American English, and it shows 'one-on-one' with approximately a 3:1 ratio over 'one-to-one'.
However, that includes all usages of 'one-on-one' and does not distinguish between the confrontation meaning versus the face to face meaning. I'm not sure how to make it distinguish meaning, but looking at context results, the two uses of 'one-on-one' seem to be about equal.
Ergo, I'm guessing that the 'face-to-face' meaning of 'one-on-one' and 'one-to-one- are much closer, but still favoring one-on-one.
Freq. results for 'one-on-one' = 1891 (both meanings)
Freq. results for 'one-to-one' = 484
So if roughly 1/2 (just a quick visual estimate, could well be higher or lower) of the one-on-one contexts represent the face-to-face aspect, then the two are closer, but still favor one-on-one.
To clarify, by face-to-face, I probably should have said something like personal 1:1 meetings. It wouldn't literally have to be in person.